Evaluating the CDF of the Skew Normal Distribution

Christine Amsler, Michigan State University

Alecos Papadopoulos, Athens University of Economics and Business ∗

Peter Schmidt, Michigan State University April 3, 2020

Final version

Abstract

In this paper we consider various methods of evaluating the cdf of the Skew Nor- mal distribution. This distribution arises in the stochastic frontier model because it is the distribution of the composed error, which is the sum (or difference) of a Normal and a Half Normal random variable. The cdf must be evaluated in models in which the composed error is linked to other errors using a Copula, in some methods of goodness of fit testing, or in the likelihood of models with sample selection bias. We investigate the accuracy of the evaluation of the cdf using expressions based on the bivariate Normal distribution, and also using simulation methods and some approximations. We find that the expressions based on the bivariate Normal distribution are quite accurate in the central portion of the distribution, and we propose several new approximations that are accurate in the extreme tails. By a simulated example we show that the use of approximations instead of the theoretical exact expressions may be critical in obtaining meaningful and valid estimation results.

Keywords: skew normal distribution, bivariate normal distribution, stochastic frontier, simulation, computational software.

JEL classiﬁcation: C46, C87, C88, C13.

∗Corresponding author, e-mail: [email protected], ORCID ID: 0000-0003-2441-4550.

1 1 Introduction

This article deals with the evaluation of the cumulative distribution function of the Skew Normal distribution (hereafter SN-cdf). This is the distribution of the Normal / Half- Normal composed error in stochastic frontier (SF) models, and that is our main interest. However, the Skew Normal distribution also arises in a number of other contexts. For example, the maximum and the minimum of two correlated bivariate standard Normal random variables follow Skew Normal distributions (Loperfido, 2002). As another example, if X,Y are correlated bivariate Normal variables, then the distribution of X Y > 0 is Skew | Normal (Azzalini and Capitanio, 2014, p. 28). As a third example, in the Normal / Half Normal specification of the two-tier stochastic frontier model, the composed error is made up of three components and its density includes the difference of two Skew Normal cdfs (Papadopoulos, 2015). As a final more general example, the Skew Normal distribution is a useful way to model a moderately skewed regression error (which is often the case), and if one wants to account for sample selection bias, the SN-cdf will appear in the likelihood. Our work is an extension of Amsler et al. (2019) –hereafter AST– who addressed the same problem but who considered a much more limited set of methods of evaluation of the SN-cdf than we do in this paper. They proposed a simulation-based method that is accurate in the lower tail, and an approximation due to Tsay et al. (2013) that is accurate in the upper tail. For some uses, such as testing goodness of fit using the Kolmogorov-Smirnov test, the tails are not important. However, for other uses they are important. For example, suppose that a Skew Normal random variable is linked to other random variables using a Copula such as the Gaussian Copula. Then the SN-cdf is an argument of the inverse of the standard Normal cdf. This will explode to infinity (or minus infinity) if the argument is exactly one (or zero), causing the estimation algorithm to break down. So it is important that the evaluations of the SN-cdf are not exactly zero or one, even unreasonably far into the tails. The methods considered by AST achieve this, but it is not clear how accurate they are in the central portion of the distribution, which is presumably where most of the observations will lie. We will consider two other simulation-based methods, two other upper tail and two other lower tail approximations, and a variety of methods of evaluation of closed-form expressions

2 based on the cdf of the bivariate Normal distribution. The numerically evaluated closed- form expressions turn out to be quite accurate in the central portion of the distribution, where “central portion” is defined rather liberally as the range where the cdf value is greater than 10−20 and less than 1 10−20. Our lower tail approximations are also quite accurate − in the extreme lower tail (cdf value less than 10−20 but large enough to calculate, which, given standard floating-point double-precision arithmetic, means larger than about 10−310). Similarly, our upper tail approximations are quite accurate in the extreme upper tail (cdf value greater than 1 10−20, but small enough to calculate, which means smaller than − about 1 10−310). − The plan of the paper is as follows. Section 2 will define notation and present some preliminary results. Sections 3, 4 and 5 will present the various methods of evaluation of the SN-cdf that we will consider. Section 6 will present and discuss the results of our evaluations. In section 7 we provide a simulated example to show that accurate evaluation deep into the tails of the SN-cdf and the use of approximate expressions may be necessary to obtain meaningful and valid estimation results. Finally, Section 8 will give our conclusions. An Appendix and an on-line supplementary file complete the work.

2 Preliminaries

We will follow the AST notation. We consider the Skew Normal distribution with location parameter equal to zero; scale parameter σ > 0; and shape parameter λ R. This ∈ distribution is characterized by the Skew Normal density snλ,σ (ε)=(2/σ)φ(ε/σ)Φ(λε/σ), where φ and Φ are the standard Normal pdf and cdf respectively. We want to calculate the Skew Normal cdf P (Q)= P (ε Q)= Q sn (ε) dε. λ,σ ≤ −∞ λ,σ More speciﬁcally, we will consider the caseR of σ = 1 and therefore investigate the accuracy of the evaluation of Pλ,1 (Q), which we will write more simply as Pλ(Q). This does not entail any loss of generality, because for σ = 1, one can use the fact that P (Q)= 6 λ,σ P (Q/σ). Similarly, we will consider only the case of λ 0. For λ < 0 we can use the λ ≥ so-called “reﬂection property” of the distribution, that P− (Q)=1 P ( Q). λ − λ − The connection of the Skew Normal distribution to the SF model is as follows. The “composed error” is ε = v + u, where v N (0, σ2), u N +(0, σ2), and v and u are ∼ v ∼ u 3 2 2 2 independent. Let σ = σu + σv and λ = σu/σv. Then ε has the Skew Normal density snλ,σ (ε). This discussion is for the case of ε = v + u, which would be natural in a cost frontier, and follows the discussion in Tsay et al. (2013) and AST. In the case of a production frontier, as in the original papers of Aigner, Lovell, and Schmidt (1977) and

Meeusen and van den Broeck (1977), we would want to consider ε∗ = v u. Then ε∗ has − density sn (ε∗)=(2/σ)φ(ε∗/σ)Φ( λε∗/σ); that is, we just change the sign of λ. λ,σ − AST evaluated the cdf Pλ(Q) by a simulation based on the representation

P (Q)= E (Φ[(Q u)/σ ]) . (1) λ u − v Here σ2 =1/ (1 + λ2), σ2 =1 σ2 = λ2/ (1 + λ2), u N(0, σ2)+, and E represents the v u − v ∼ u u expected value over the Half-Normal distribution of u. They estimated this by averaging Φ[(Q u)/σ ] over a large number of draws from the distribution of u. Speciﬁcally, they − v used 10,000,000 draws and got results that they regarded as reliable for negative values and small positive values of Q, but not for large positive values of Q. They created a large tabulation of the cdf as a function of λ and Q and suggested interpolation in this table. However, many people would probably prefer a simple calculation that is easier to program than an interpolation, which we will provide. AST also considered an approximation due to Tsay et al. (2013), p. 261, equations (12) and (13). AST concluded that this approximation is accurate for large positive values of Q but not for negative values.1

For the special case of λ = 1, AST also provided an “exact” result, namely P1 (Q) = [Φ (Q)]2.2 They were unaware that this result was already known.3 This result can be regarded as exact because it is not an approximation and it can be calculated very accurately over an extremely wide range of Q. It is useful as a check on the accuracy of the other methods of evaluating the SN-cdf.4 1Ashour and Abdel-hameed (2010) presented also a closed-form approximation to the Skew Normal density and CDF, but they were functions with branches while the partition of the support depended on the value of λ. They considered only the [-4,4] interval. 2This is the cdf of the maximum of two i.i.d. standard Normal variables, and can also be obtained as the limiting case of the results in Loperﬁdo (2002) mentioned earlier. 3Azzalini (1985), p.174. This paper by Adelchi Azzalini has become by all accounts a classic of the statistical literature, and so, the Mark Twain remark on classic works applies. 4Gupta and Chen (2001) used it also as a validation tool for their SN-cdf tables. They applied Simpson’s

4 Our aim is to provide reliable and simply calculated alternatives to the methods of AST and Tsay et al. (2013) for the lower and upper tails, respectively, and also to provide alternatives that can handle the central portion of the distribution reliably.

3 Expressions based on the bivariate Normal distribution function

Azzalini (1985) expressed Pλ(Q) using Owen’s T -function T (h, a), presented and tabulated in Owen (1956), as follows:

P (Q)=Φ(Q) 2T (Q, λ) . (2) λ −

Since Owen devised his function in order to provide a computational formula for probabilities related to the bivariate Normal distribution, it is not surprising that Pλ(Q) can also be expressed in terms of the standard bivariate Normal cdf Φ2:

2 P (Q)=2Φ2(Q, 0; ρ), ρ = λ/√1+ λ . (3) λ − This formula can be found in Azzalini and Capitanio (2014) -hereafter AZC-, p. 34, eq. (2.34). An analytical proof is given by Papadopoulos (2018), p. 286-288. We present a very simple proof in Appendix A of this paper. Using the symmetries of the bivariate standard Normal integral when one of the variables is ﬁxed at zero, alternative expressions to equation (3) are:

P (Q) = 2Φ(Q) 2Φ2(Q, 0; ρ) (4) λ − −

P (Q)=1 2Φ2( Q, 0; ρ) (5) λ − − − While theoretically equivalent, these alternative formulas involve subtraction, which means that computational issues may make them produce diﬀerent results especially in the tails. rule to compute the integral, and examined values in the [ 4, 4] interval. −

5 One could argue philosophically about whether expressions involving Φ2 are “closed form” expressions.5 What is important to take into consideration is that the bivariate standard Normal cdf appears as a ready-made “special function” in almost all computing, statistical, and econometric software. Therefore, it is highly likely that it will be the method most often used to compute the SN-cdf, making it a prime target of study. The history of methods to compute the multivariate Normal integral is long. Related material can be found in Tong (1990), ch. 8, and in Genz and Bretz (2009), which identify the first work on the matter as far back as 1858, and also note that the method of Owen (1956) was used for decades as the preferred one to compute bivariate Normal probabilities. It appears though that modern day software prefers other algorithms: for example the Matlab “mvncdf” function uses for the bivariate case (quote) “adaptive quadrature on a transformation of the (Student’s)-t density, based on methods developed by Drezner and Wesolowsky (1989) and by Genz (2004)”.6 The absolute value of the correlation coefficient appears to matter, and not in the same way for each approximation method. For example, in Mee and Owen (1983) the approximation error increases visibly in relative terms if ρ > 0.5, and even more so | | when correlation exceeds 0.7. On the other hand, the methods mentioned in the previous paragraph are at their best when correlation is closer to unity (see Genz, 2004). This has relevance to our task here, because in the stochastic frontier framework this correlation coefficient is a strictly increasing function of the parameter λ: already for λ = 0.6 the implied ρ in the Φ2 expression for the SN-cdf takes the value ρ = 0.514. For λ = 1 we | | have ρ =0.707. For λ = 2 the corresponding value jumps to ρ =0.88. Now, the signal- | | | | to-noise ratio (SNR) in Normal-Half Normal SF models (ratio of standard deviations of inefficiency over noise) is SNR = 1 2/π λ 0.6λ.7 So we will have a correlation − · ≈ coefficient higher than 0.5 fora SNRp as low as 0.36. Encountering real-world samples with, say, λ =2 = SNR = 1.2 is totally within reason. So in general, we anticipate that ⇒ 5The fact that one of the two arguments is fixed at zero helps the affirmative argument in our case. 6MATLAB on-line documentation, https://www.mathworks.com/help/stats/mvncdf.html. Website ac- cessed July 6th, 2019. 7Some authors define the “signal to noise” ratio to be the ratio of the variance of the inefficiency component over the variance of the composed error. This is a useful metric but it is not a signal-to-noise ratio, as the concept is used in most scientific fields.

6 in applied studies we will have to deal with a high correlation coeﬃcient in the bivariate Normal integral representation.

We will use a variety of different software, commercial or open source, for evaluating Φ2, including Matlab, Mathematica, Casio-Keisan and two different routines in R, one created by A. Azzalini (“sn” package) and the other by A. Genz and associates (“mvtnorm” package). Also, the software Mathematica has a built-in “Skew Normal cdf” command, which uses expression (2).8 These different routines do not necessarily give the same answers, especially in the tails.

4 Alternative expressions for evaluating the SN-cdf via simulation

As noted in Section 2, AST evaluated the cdf Pλ(Q) by a simulation based on the representation P (Q)= E Φ[(Q u)/σ ], u N(0, σ2)+, where E represents the expectation over λ u − v ∼ u u the distribution of u. We can apply the same approach starting from the Φ2 expression, and we show in Appendix B that the SN-cdf can be written as 1 Q P (Q)= σ E exp λ2z2 Φ λz , (6) λ z z 2 σ − z where z N(0, σ2)+ and σ2 = 1/(1 + λ2) = σ2. This would be evaluated by averaging ∼ z z u the expression σ exp λ2z2/2 Φ(Q/σ λz) over draws from the distribution of z. Asa z { } z − theoretical result, this is not too interesting because it is equivalent to the much simpler expected value in the AST paper. However, since simulation is involved, having a second expression is a way to check and validate the whole approach, by computing both and comparing the results, especially in the extreme tails and/or for extreme values of λ. Another simulation-based method of evaluating the cdf can be derived from a result in Tong (1990). We show in Appendix C that, starting from Tong’s equation (2.2.3) p.15, we can derive the following expression:

8Casio-Keisan is an on-line general computing tool, https://keisan.casio.com. It uses a proprietary method of “high-precision” arithmetic, which allows for computational accuracy far beyond that of the ﬂoating-point, double-precision arithmetic widely implemented in most available hardware and software. The Mathematica software also uses a high-precision proprietary method.

7 w ρ Q w ρ Pλ (Q)=2Ew Φ | | Φ − | | , (7) " 1 ρ ! · 1 ρ !# p−| | −|p | where ρ = λ/√1+ λ2 and w N(0, 1).p This expressionp can also be evaluated by simula- | | ∼ tion, averaging the expression over draws from the distribution of w.

5 Some useful approximations for the tails

5.1 Upper tail

AST conluded that the Tsay et al. (2013) approximation is very accurate in the upper tail. A relevant numerical observation is that the Tsay approximation is essentially equal to 1 2Φ( Q) in the extreme upper tail (cdf values greater than 1 10−20). It is interesting − − − that this does not depend on λ. Formally, this leads to an upper tail approximation (which we will label APS-UT),

Pλ(Q) = 1 2Φ( Q) = 2Φ(Q) 1 . (8) ∼ − − − The 1st derivative of this expression is 2φ(Q) > 0 and so the monotonicity property is satisﬁed, which is a minimum requirement for a function to be a valid approximation for a CDF. In this particular case, APS-UT is a distribution function on its own for positive values, that of the Half-Normal distribution. And in fact, our interest in this expression was originally as an approximation to the SN-cdf for large λ, since as λ , the distribution →∞ of ε approaches the Half-Normal, N(0, 1)+. But it turns out that λ does not matter when Q is very large. A numerical explanation of why this is the case starts with eq.(4),

P (Q) = 2Φ(Q) 2Φ2(Q, 0; ρ). From this expression it follows that λ − −

1 P (Q)=2[1 Φ(Q)] + 2 [Φ2(Q, 0; ρ) 1/2] . (9) − λ − − −

As Q , Φ2 (Q, 0; ρ) 1/2 much faster than Φ(Q) 1. A more precise theoretical →∞ − → → justiﬁcation for our approximation is as follows. The ﬁrst result in Proposition 2.8 of Azzalini and Capitanio (2014), p. 35, says that (for λ> 0)

[1 Pλ (Q)] lim →∞ − =1 . (10) Q [2φ(Q)/Q]

8 This leads to another large-Q approximation (AZC-UT), 2φ (Q) P (Q) = 1 . (11) λ ∼ − Q One can compute that the 1st derivative of this expression is everywhere positive, so it can be a valid cdf approximation. For the expression to be equivalent in the limit to the APS-UT approximation, we require, as Q , that φ(Q)/[QΦ( Q)] 1. This amounts →∞ − → to h(Q)/Q 1 where h (Q)= φ(Q)/Φ( Q) is the Normal hazard function, and it is well → − known that h(Q)/Q 1 as Q . So in the limit our expression and the expression → → ∞ based on Azzalini and Capitanio (2014) are the same. (As noted above, neither of these depends on λ).

5.2 Lower tail

We obtained the following lower tail approximation, labeled APS-LT, by examination of AST’s calculated cdf values in the lower tail:

2 Pλ (Q) ∼= 2Φ(Q)Φ(λQ) /(1 + λ ) . (12) For positive λ this formula is evidently monotonic in Q. The approximation is exact for λ = 1, Q. For other values of λ, it is exact in the limit, in the sense that the limit of the ∀ left hand side divided by the right hand side approaches one as Q . To show this, → −∞ we use the second result in the aforementioned Proposition 2.8 of Azzalini and Capitanio (2014). This result says that, for α< 0,

[1 P (x)] 2 φ x√1+ α2 x = − α 1, q(x, α)= . (13) →∞ ⇒ q(x, α) → π α (1+ α2)x2 r | | The result is for negative α and large positive x. We are interested in positive λ and large in absolute value negative Q, so we let Q = x and λ = α, and we use the reﬂection − − property to obtain a second lower-tail approximation (AZC-LT),

2 φ Q√1+ λ2 Pλ(Q)=1 P−λ( Q) ∼= q( Q, λ)= 2 2 . (14) − − − − rπ λ(1 + λ )Q For Q< 0,λ> 0, this is a monotonic function of Q. To show that our approximation, eq.(12), is also exact in the limit, we need to show that the limit (as Q ) of the → −∞ 9 right hand side of eq.(14) divided by the right hand side of (12) equals one. We show this in Appendix D, where we also give a more complicated but more fundamental proof of the validity of our approximation. In a nutshell, we evaluate the derivative with respect to Q of Pλ(Q) and of our approximation, and we show that the limit of their ratio equals one.

By l’ Hôpital’s rule, this implies that the limit of the ratio of Pλ(Q) to our approximation also equals one. For finite values, this ratio is equal to one at first-order approximation, and deviations from this value lie two orders of magnitude below, and also tend to offset each other. This approach also allows to correctly predict the direction of the bias of the approximation: specifically we show that for λ > 1 the APS-LT approximation will underestimate the true value of the SN-cdf, while if λ < 1, APS-LT will overestimate the true value. This is verified by the tables.

6 Presenting and comparing evaluations of the cdf

In attempting to compute the value of a cdf far into the tails, one pervasive issue of numerical accuracy is to keep the cdf value from rounding to exactly one in the upper tail. In comparison, rounding down to zero is a much less severe problem. This is no accident: due to the way “floating-point” arithmetic constructs its numbers (e.g. the ubiquitous IEEE-754 binary 64 format standard), zero is a much “thinner” computational target than unity: by direct computation we can get to a distance of 10−310 from zero, but only to ≈ a distance of 10−16 from unity. ≈ Capitanio (2010) has shown that the rate of decay of the tail probabilities is different between the two tails of the Skew Normal density. Specifically for the positively skewed case that we are examining, the (long) right tail probability decays (approaches zero) at the same rate as a Normal random variable, but the left-tail probability approaches zero faster. This is consistent with the general result that the Skew Normal distribution cannot have fatter tails in its density than the Normal, since, as regards its associated Extreme Value distribution, it belongs to the same domain of attraction (of the Gumbel distribution) as the Normal (see Chang and Genton, 2007). So the positively skewed Pλ (Q) goes to zero faster than it goes to unity, and it goes to unity at the same rate as the Normal cdf. If we consider the negatively skewed case, the (long) left tail probability decays at the same

10 rate as the Normal, while the right tail probability decays faster. So the negatively skewed

P−λ (Q) goes to unity faster than it goes to zero, and it goes to zero at the same rate as the Normal cdf. Combining this with the previously mentioned asymmetry in the computational accuracy near zero and near unity, we conclude that the furthest and most accurate we can go is towards the long left tail of the negatively skewed distribution.9 Then, using the reﬂection property we can provide values for the upper tail of the positively skewed distribution as “one minus...”. For example, consider the case of λ = 1, Q = 12. The Mathematica Skew

Normal cdf command gives P1 (12) = 1 (exactly), and the same problem would arise with all of our various evaluations of Φ2(12, 0; 0.717017). This is not a useful result. But using − the reflection property, we write P1 (12) = 1 x where x = P−1( 12) = 3.55296e-33, which − − is a meaningful and reliable calculation, and so we report that P1 (12) = 1 3.55296e-33. − For reasons of uniformity, we report the SN-cdf values in this format for Q 2 in all tables. ≥ We turn now to the Skew Normal cdf evaluation results, which are contained in Tables 1 - 6 and their subtables. Each table gives results for a different value of λ, ranging from λ =0.25 to λ = 8. We consider values of Q that range from -28 to 32. The entries in the tables are the cdf values. The key “***” indicates that the evaluation was exactly zero or exactly one; “DNA” indicates a case that does not apply (e.g. a lower tail approximation for Q > 0); and “NEG” indicates that the result was a negative number. Some of the tables are to be found in the on-line supplement to this article. We considered 17 different methods of evaluation. Tabulated, these are:

9Using the negatively skewed distribution, we can obtain non-zero values for Q = 38, that are of order − 10−316, where we hit the limits of double-precision arithmetic.

11 Evaluation methods.

Key Expression and Software/routine used Simulations, 10,000,000 draws 1. AST eq.(1), Matlab 2. TONG eq.(7), Matlab 3. APS-SIM eq.(6), Matlab Approximations 4. TSAY Tsay et al. (2013), Q R , Matlab ∈ 5. APS-LT eq.(12), Q< 0, Matlab 6. AZC-LT eq.(14), Q< 0, Matlab 7. APS-UT eq.(8), Q> 0, Matlab 8. AZC-UT eq.(11), Q> 0, Matlab

Exact eq.(3) 2Φ2(Q, 0; ρ)

9. Φ2-MTL Matlab-mvncdf

10. Φ2-R-MVTN R-mvtnorm

11. Φ2-R-SN R-sn

12. Φ2-MTH Mathematica CDF[Binormal]

13. Φ2-CK Casio-Keisan Other evaluations, based on subtraction 14. MATH-SN eq.(2), Mathematica CDF[SkewNormalDistribution] 15. OWEN-T eq.(2), R-sn

16. 2(Φ Φ2) eq.(4), R-mvtnorm − 17. 1 2Φ2 eq.(5), R-mvtnorm −

6.1 The case of λ =1.

We will begin with the case of λ = 1, because here we have the exact standard of compar- 2 ison, Pλ(Q) = [Φ(Q)] , which we also present. The results are given in Table 1, with four sub-tables 1a, 1b, 1c and 1d. Table 1a gives the exact results and also the results of the three simulation-based evaluations. None of these is reliable for positive Q. But AST and APS-SIM are essentially the same and are quite accurate for Q 0, even into the extreme lower tail. For example, for ≤ Q = 24, AST, APS-SIM and the “exact” [Φ(Q)]2 all yield 1.93e-254. TONG fails in the − extreme lower tail. Simulation-based methods are perhaps too cumbersome for practical

12 use, but we are interested in them as a standard of comparison for lower tail results for other values of λ, for which “exact” results are not available. Table 1b gives the results for our various approximations. As in the AST paper, TSAY is accurate for Q 0 but not for negative Q. APS-LT is exact (in both tails) when λ = 1. ≥ AZC-LT is accurate in the lower tail (cdf values 10−20) but not for less extreme negative ≤ values of Q. APS-UT is accurate for Q 4, including the extreme upper tail, but not for ≥ smaller positive values of Q. AZC-UT is similar to APS-UT, so it is accurate in the upper tail, but it is not quite as good as APS-UT.

Table 1c gives the results for five different evaluations of 2Φ2(Q, 0; ρ). All of these are quite accurate in the center of the distribution (10−20 cdf value 1 10−20). None of ≤ ≤ − them is reliable in the lower tail. Except for Casio-Keisan, they are quite accurate in the upper tail. The problem with Casio-Keisan is not inaccuracy in its evaluations but just that it rounds to exactly one in the extreme tails. Table 1d gives the results for our other methods. We will group these as the “methods that involve subtraction” and subtraction might give results of questionable validity when the differences are very small, due to limits in digit-accuracy. Nevertheless, MATH-SN (the Mathematica built-in Skew Normal cdf calculator) is accurate across the whole range of the distribution. 2(Φ Φ2) is accurate in the central portion of the distribution and in the − upper tail. 1 2Φ2 is not as good, largely because it fails to evaluate in the tails or even − closer to the center. OWEN-T is very bad.

[TABLES 1a, 1b, 1c, 1d SOMEWHERE HERE]

6.2 Other values of λ.

Based on the previous results, we disqualiﬁed TONG, Φ2-CK, OWEN-T and 1 2Φ2 from − further consideration. The results for these four methods for other values of λ are available as a supplement to this paper. The results for the other methods are given in Tables 2 to 5, (each consisting of two sub-tables), for λ =0.2, 0.5, 2, 4. We will discuss the results for the central portion of the distribution and for the upper and lower tails separately. Much of this discussion is very similar to the case of λ = 1.

13 6.2.1 The central portion of the distribution (10−20 cdf value 1 10−20) ≤ ≤ −

All of the evaluations of Φ2 give more or less the same answers in this range and are therefore presumed to be accurate. Also they agree quite closely with the AST results for negative Q, which is another indication that they are accurate. The methods that involve subtraction also agree, except that for large values of λ they break down (return negative probabilities) in the lower end of this range. The lower tail approximations are not as accurate as the Φ2 evaluations for negative Q in this range of probabilities. The upper tail approximations APS-UT and AZC-UT are not very accurate for positive Q in this range of probabilities, except when λ is large. TSAY is much better.

6.2.2 The lower tail (cdf value < 10−20)

The upper tail approximations do not apply and TSAY is not accurate in the lower tail. The simulated methods (AST and APS-SIM) and the two lower tail approximations (APS-LT and AZC-LT) agree quite well, except for some minor discrepancies for the largest values of λ. We conclude that they are accurate. The various evaluations of 2Φ2 agree reasonably well with each other, but not with the simulations or lower tail approximations. We don’t trust them because we found exactly the same pattern when λ = 1, that is, they matched each other but did not match the “exact” [Φ(Q)]2 results in the lower tail (whereas AST, APS-SIM, APS-LT and AZC-LT did match the “exact” results).

6.2.3 The upper tail (cdf value > 1 10−20) − The lower tail approximations do not apply and the simulation-based evaluations are not accurate. TSAY, APS-UT and AZC-UT agree in the upper tail and we conclude that they are accurate. The various Φ2 evaluations and the methods that involve subtraction also agree with each other and with TSAY, APS-UT and AZC-UT, so we conclude that they are also accurate.

[TABLES 2 to 5 SOMEWHERE HERE]

14 7 Usefulness of the approximations: An example with simulated data

To dispel any notion that the focus on the accurate evaluation of the extreme tails of the SN-cdf has no practical relevance, we present a short example with simulated data to show how it may become a critical matter. We consider the following data-generating mechanism, a cost stochastic frontier:

∗ ′ ′ y = x β + ε, x = (1, x) , β =(β0, β1) + x χ2, ε = v + u, v N 0, σ2 , u N 0, σ2 (15) ∼ 1 ∼ v ∼ u We then apply a censoring of the sample: we assume that the observable dependent variable is

y =0 y∗ 0 ≤  ∗ ∗ y = y y > 0.

The log-likelihood for such a sample is

′ ′ = ln fε(yi xiβ) + ln Fε( xiβ), L y>0 − y=0 − X X where f( ) and F ( ) are the density and distribution functions respectively. · · We will estimate three alternative expressions of this log-likelihood. In the ﬁrst, we use the Φ2 expression for the SN-cdf for all observations. In the second, we guard against zero-values for the Φ2 expression by imposing a strictly positive ﬂoor on its value, which is a traditional, handmade way to sidestep such computational hurdles. For the third ′ alternative we use Φ2 but also the APS-LT approximation whenever x β 1. After − i ≤ − discarding constants, we have

15 ′ ′ 1 = n ln s + ln φ ((yi xiβ)/s) + ln Φ (λ(yi xiβ)/s) L − y>0 − − X h i ′ λ + ln Φ2 x β/s, 0 ρ = − , (16) i √ 2 y=0 − 1+ λ X 2 2 2 with s = σv + σu, λ = σu/σv.

The log-likelihood that uses the ﬂoor value was speciﬁed as

′ ′ 2 = n ln s + ln φ ((yi xiβ)/s) + ln Φ (λ(yi xiβ)/s) L − y>0 − − X h i ′ λ −24 + max ln Φ2 x β/s, 0; ρ = − , ln(10 ) . (17) − i √ 2 y=0 1+ λ X n o The ﬂoor value of 10−24 is “virtually zero” so it would appear to shield against exact zero values without aﬀecting the iterative optimization algorithm. We note that for scripting purposes, our advice would be to replace the max operator with indicator functions, since maximum likelihood will use derivatives (exact or numerical) in its optimization algorithm. Finally, the log-likelihood that uses the APS-LT approximation is

′ ′ 3 = n ln s + ln φ ((yi xiβ)/s) + ln Φ (λ(yi xiβ)/s) L − y>0 − − X h i ′ λ + ln Φ2 x β/s, 0; ρ = − − i √ 2 =0 −x′ −1 1+ λ y , Xiβ> + ln(1 + λ2) + ln Φ ( x′ β/s) + ln Φ ( λx′ β/s) . (18) − − i − i =0 −x′ ≤−1 y , Xiβ h i 7.1 Simulation and estimation results

For this example we used the open-source software G.R.E.T.L.10, or simply gretl (and “hansl” indeed, which is the name of the scripting language of gretl), a software that is dedicated to econometrics rather than to computation or statistics in general. Gretl uses an SMID-oriented Fast Mersenne twister to generate pseudo-random numbers whose quality

10“Gnu Regression, Econometrics and Time-series Library, http://gretl.sourceforge.net/.

16 has been veriﬁed by Yalta and Schreiber (2012). To compute the bivariate Normal integral it uses essentially the same algorithms as the “mvncdf” function in Matlab.11

We generated ﬁve samples of n = 1000, setting β0 = 1, β1 = 1 , σv = 1 , σu = 0.5. The number of negative y∗ observations ranged from 48 to 83 and these were censored to zero to obtain the y variable. Maximum likelihood estimation was carried out using the widely adopted algorithm of Broyden, Fletcher, Goldfarb and Shanno (BFGS). For each sample, we tried three diﬀerent sets of starting values, one of which contains the true values of the parameters. We present here the results from sample #1. We discuss also the other samples, whose detailed results are included in the Supplement to this article.

Estimation results under alternative likelihood expressions. Simulated data. true values 1 1 1 0.5 Sample #1 OLS estimates 1.47 0.96 n=1000 True error sample skewness 0.10 censored obs : 61 OLS residuals skewness 0.27 Starting log-likelihood values set b0 b1 s v s u Notes Starting values OLS OLS 1.2 0.6 F only 1.45 0.96 0.96 0.00 non-zero gradient - std errors not computable 1 2 F2 with floor value 1.45 0.96 0.96 0.00 non-zero gradient - std errors not computable

F2 / APS-LT 1.03 0.97 0.97 0.52 Starting values OLS OLS 0.6 1.2 F only n/a n/a n/a n/a zero values of F due to starting values 2 2 2 F2 with floor value 1.45 0.96 0.96 0.00 non-zero gradient - std errors not computable

F2 / APS-LT 1.03 0.97 0.97 0.52 Starting values 1 1 1 0.5 F only 1.46 0.96 0.96 0.00 non-zero gradient - std errors not computable 3 2 F2 with floor value 1.46 0.96 0.96 0.00 non-zero gradient - std errors not computable

F2 / APS-LT 1.03 0.97 0.97 0.52

Estimation with the log-likelihoods that use the Φ2 expression of the SN-cdf for all observations is clearly problematic: without the ﬂoor value, the iterative algorithm may be stopped because we may get exact zero-values already due to the starting values. When this does not happen it produces bad results: essentially it cannot escape the initial starting values for the regression coeﬃcients and it attempts to optimize the likelihood by sending

11For the 132 values per computing algorithm that we computed in our tables, the two algorithms diﬀered only in 5 left-tail cases for λ 2, for which gretl computed exactly zero while Matlab gave a non-zero ≥ value. On the other hand, gretl is more tidy in that, where Matlab gave negative values, gretl gave zeros.

17 the estimate of the σu parameter to zero. Note that this could also happen if we had a case of “wrong skew” (negative in our case, since the one-sided component of the composed error term enters positively). But we don’t, since both the true error sample skewness as well as the OLS residual skewness is estimated as positive. When it exits, the gradient of the likelihood is far from zero, and standard errors are not computed, even when using the Outer Product matrix of the contributions to the gradient. The problem is that the matrix is non-invertible (it is positive semi-definite by construction). Imposing a “virtually zero” floor value doesn’t fare better. Here, the detailed iterations reveal that the algorithm initially moves away from the OLS starting values, only to eventually return. In contrast, the likelihood that uses the APS-LT approximation for the lower tail of the SN-cdf does fine in all cases. So it is not just a matter of avoiding exact-zero values. The accuracy of the evaluation of the SN-cdf is critical here.

The picture for the two likelihoods that use only the Φ2 expression remains the same in the other four samples that we examined: in all cases the regression coeﬃcient estimates remain stuck at the OLS values, the σu parameter is estimated as zero, and, in almost all cases standard errors cannot be computed. As regards the likelihood that uses the APS-LT approximation, in two of the other four samples it performed as above, while in the other two, it did not manage to converge in the sense that that norm of the gradient was too high, but still at the point of exit, it gave very accurate estimates for the four unknown parameters.

8 Conclusions

The reader can draw his or her own conclusions from our tables and simulations, but our recommendation would be the following. In the central portion of the distribution, use Φ2(Q, 0; ρ) applying any of the standard evaluation methods we discussed, or use the Mathematica built-in Skew Normal function. In the lower tail, use one of the two lower tail approximations that we have proposed (APS-LT or AZC-LT). In the upper tail, continue to use Φ2(Q, 0; ρ) or the Mathematica function, or else use one of the upper tail approximations

(TSAY, APS-UT or AZC-UT). The approximations are numerically simpler than the Φ2 evaluations, but from a programming perspective it may be easier to just use the Φ2

18 evaluation everywhere except in the lower tail: the simulation exercise earlier showed that accurate estimation deep into the lower tail may be crucial in obtaining meaningful results. Moreover, seeing approximate expressions perform better in the short and hence more diﬃcult tail of the distribution than the exact theoretical expressions, is a strong example for the fact that applied computational technology is a gatekeeper for valid statistical inference that we cannot ignore.

Compliance with ethical standards

Conﬂict of Interest: All authors declare that they have no conﬂicts of interest. Ethical Approval: This article does not contain any studies with human participants or animals performed by any of the authors.

Appendix

A. Proof of eq. (3)

Let X,Y be standard bivariate Normal with correlation δ = λ/√1+ λ2. Then Z = X Y > | 0 is Skew Normal with scale parameter equal to one and shape λ (Azzalini and Capitanio, 2014, p. 28). So P (Q)= P (Z Q)= P (X Q Y > 0) λ ≤ ≤ |

P (X Q,Y > 0; δ) = ≤ =2P (X Q,Y > 0; δ) P (Y > 0) ≤

2 =2P (X Q,Y < 0; δ)=2Φ2(Q, 0; ρ), δ = ρ = λ/√1+ λ . ≤ − − −

B. Derivation of eq.(6)

Writing out explicitly the Φ2 expression for Pλ(Q), we have

0 Q 1 z2 x2 ρz P (Q)=2 exp + x dxdz. λ 2 2 2 2 −∞ −∞ 2π 1 ρ −2(1 ρ ) − 2 (1 ρ ) 1 ρ Z Z − − − − p 19 Using the relation ρ = λ/√1+ λ2 we can re-write this as −

0 Q √1+ λ2 1 1 P (Q)=2 exp 1+ λ2 z2 1+ λ2 x2 λ√1+ λ2zx dxdz, λ 2π −2 − 2 − Z−∞ Z−∞ and we want to manipulate this into an expected value from a known distribution. We can write it as

Pλ (Q)= 1 0 √1+ λ2 1 Q 1 = 2 exp 1+ λ2 z2 exp 1+ λ2 x2 λ√1+ λ2zx dxdz. √2π −∞ √2π −2 −∞ −2 − Z Z The integrand in the outer integral is now a Normal density truncat ed at ( , 0]. −∞ Swapping the integration limits of the outer integral we have

Q 1 1 2 2 2 Pλ (Q)= Ez exp 1+ λ x + λ√1+ λ zx dx z 0 , √2π −∞ −2 ≥ Z z N 0, σ2 . ∼ z 2 where σz = 1/√1+ λ . So z can be obtained as a zero-mean Normal random variable truncated from below at zero, i.e a Half-Normal. To compute the remaining single integral we use the identity

Q ∞ ∞ = . − Z−∞ Z−∞ ZQ Now, from Gradshteyn and Ryzhik (2007), formula 3.323(2) p. 337, we have

∞ 1 exp 1+ λ2 x2 + λ√1+ λ2zx dx −2 Z−∞

√2π λ2 (1 + λ2) z2 √2π λ2 = exp = exp z2 . √ λ2 2(1+ λ2) √ λ2 2 1+ 1+ Next, using formula 3.322(1) p. 336 from the same source (while noting that the authors use the symbol Φ to denote the error function and not the standard Normal cdf as we do here), we have

20 ∞ 1 exp 1+ λ2 x2 + λ√1+ λ2zx dx −2 ZQ

√π λ2 (1 + λ2) z2 λ√1+ λ2 Q√1+ λ2 = · 2 exp 2 erfc 2 z + √2 √1+ λ 2(1+ λ ) · −√2 √1+ λ √2 ! · ·

√π λ2z2 = · exp 2 1 Φ λz + Q√1+ λ2 √2 √1+ λ2 2 · · − − · h i

√2π λ2z2 = · exp Φ λz Q√1+ λ2 . √1+ λ2 2 · − So

∞ ∞ √2π λ2 2 √ 2 = 2 exp z Φ λz + Q 1+ λ . −∞ − Q √1+ λ 2 · − Z Z

2 Inserting into the expected value expression and using σz =1/√1+ λ , we arrive at

λ2 P (Q)= σ E exp z2 Φ( λz + Q/σ ) z 0 , z N 0, σ2 . λ z z 2 · − z | ≥ ∼ z or

1 Q P (Q)= σ E exp λ2z2 Φ λz , z N(0, σ2)+. λ z z 2 σ − ∼ z z C. Derivation of eq.(7)

Tong (1990), eq. (2.2.3), p. 15, provides the following expression for the bivariate Normal distribution function, with zero mean vector and unitary variances:

∞ w ρ + x1 δ w ρ + x2 1 ρ 0 ρ ≥ Φ2(x1, x2; ρ)= φ(w) Φ | | Φ | | dw, δρ =  . −∞ 1 ρ · 1 ρ Z " p ! p !#  1 ρ< 0 −| | −| | − p p 21  For the case of the Skew Normal cdf with positive skew, we have ρ< 0 = δ = 1, ⇒ ρ − and also x1 =0, x2 = Q, arriving at

D. The APS-LT approximation, eq. (12)

D1. Equivalance at the limit

We want to show that the ratio of the right hand side of equation (14) to the right hand side of equation (12) converges to one as Q . After a few cancellations, this ratio, → −∞ “R”, is given by

1 φ Q√1+ λ2 R = 2 . √2π λQΦ(Q)Φ(λQ) A little algebra shows that

1 φ Q√1+ λ2 = φ(Q)φ(λQ) , √2π so that

φ(Q) φ(λQ) R = . QΦ(Q) · (λQ)Φ(λQ) Letting z = Q, we have −

h(z) h(λz) R = , z · λz where h (z)= φ(z)/Φ( z) is the Normal hazard function. It is well known that h(z)/z 1 − → as z . Therefore R 1 as Q . →∞ → → −∞

D2. An alternative proof and some elaboration

The derivative of the positively skewed SN-CDF is the Skew Normal density: Q d P (Q) 2φ (t)Φ(λt) dt P (Q) = 2φ (Q)Φ(λQ) . λ ≡ ⇒ dQ λ Z−∞ 22 The derivative of the APS-LT approximation eq.(12),

2 APS-LT = Φ(Q)Φ(λQ) , 1+ λ2 with respect to Q is

d 2 APS-LT = [φ (Q)Φ(λQ) + λΦ(Q) φ (λQ)] . dQ{ } 1+ λ2 Consider the ratio of the two derivatives,

2 d APS-LT /dQ 1+ 2 [φ (Q)Φ(λQ) + λΦ(Q) φ (λQ)] { } = λ dPλ (Q)/dQ 2φ (Q)Φ(λQ) 1 Φ(Q) φ (λQ) = 1+ λ . 1+ λ2 φ (Q) · Φ(λQ) Multiply and divide the ratio inside the brackets by λQ,

d APS-LT /dQ 1 λQ Φ(Q) φ (λQ) { } = 1+ λ · dP (Q)/dQ 1+ λ2 φ (Q) · (λQ) Φ(λQ) λ · 1 Q Φ(Q) φ (λQ) = 1+ λ2 · . 1+ λ2 φ (Q) · (λQ) Φ(λQ) · We examine only strictly negative values for Q. Set Q = z > 0 and denote m (z) to − be Mill’s ratio, to arrive at (the two minus signs cancel out),

d APS-LT /dQ 1 z m (z) { } = 1+ λ2 · , z = Q, w = λQ . dP (Q)/dQ 1+ λ2 w m (w) − − λ · By a well-known property of Mill’s ratio, as Q z z m (z) 1 → −∞ ⇒ → ∞ ⇒ · → and likewise for w m (w). So ·

dA/dQ 1 1 lim = 1+ λ2 =1 . Q→−∞ dP (Q)/dQ 1+ λ2 · 1 λ In words, starting from the Skew Normal density and the derivative of the APS-LT −∞ approximation are identical, so their integrals cover the same area. That is, as we move toward minus inﬁnity, the left-tail probability of the SN-cdf and of APS-LT become closer and closer.

23 Going deeper, using the three-term asymptotic expansion for Mill’s ratio,

1 1 3 1 3 1 z m (z)= z + = 1 + = z4 z2 +3 , · · z − z3 z5 − z2 z4 z4 −

1 3 1 3 1 z2 3 w m (w)= 1 + = 1 + = z4 + , · − w2 w4 − λ2z2 λ4z4 z4 − λ2 λ4 we have

d APS-LT /dQ 1+ λ2q(z,w) z m (z) z4 z2 +3 { } = , q(z,w) · − . 2 4 2 −4 dPλ(Q)/dQ 1+ λ ≡ w m (w) ≈ z (z/λ) + 3λ · − At ﬁrst order the numerator and the denominator of q(z,w) are equal and in turn the ratio of derivatives is equal to unity at that level. The numerator and denominator of q(z,w) tend to diverge from each other from the second term on, which is two orders of magnitude smaller, while also the next terms in the sequence of the asymptotic expansion alternate in sign and so they tend to oﬀset each other. With this representation, we can determine when we expect the APS-LT approximation to be higher or lower than the actual SN-cdf value. A. If λ > 1 (and especially for the lambda values we consider), the denominator of q(z,w) becomes bigger than the numerator. Then q(z,w) < 1 and it follows that the ratio of derivatives approaches unity from below: The Skew Normal density lies above the derivative of APS-LT, and this means that we expect that APS-LT will underestimate the true value. This is validated by the Tables, comparing the APS-LT approximation with the AST simulation. B. If λ< 1 then the denominator becomes smaller than the numerator, q(z,w) > 1, and the ratio of derivatives approaches unity from above: The Skew Normal density lies below the derivative of APS-LT, and this means that we expect that APS-LT will overestimate the true value. This is again validated by the Tables, comparing the APS-LT values with the AST simulation.

24 References

Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. Journal of Econometrics 6:21–37 Amsler C, Schmidt P, Tsay WJ (2019) Evaluating the cdf of the distribution of the stochastic frontier composed error. Journal of Productivity Analysis 52:29–35 Ashour S, Abdel-hameed MA (2010) Approximate skew normal distribution. Journal of Advanced Research 1(4):341–350 Azzalini A (1985) A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12(2):171–178 Azzalini A, Capitanio A (2014) The Skew-Normal and related families. Cambridge Univer- sity Press Capitanio A (2010) On the approximation of the tail probability of the scalar Skew Normal distribution. Metron 68(3):299–308 Chang SM, Genton M (2007) Extreme value distributions for the skew- symmetric family of distributions. Communications in Statistics, Theory and Methods 36(9):1705–1717 Drezner Z, Wesolowsky GO (1989) On the computation of the bivariate Normal integral. Journal of Statistical Computation and Simulation 35:101–107 Genz A (2004) Numerical computation of rectangular bivariate and trivariate Normal and t probabilities. Statistics and Computing 14(3):251–260 Genz A, Bretz F (2009) Computation of multivariate Normal and t probabilities. Springer Science and Business Media Gradshteyn IS, Ryzhik IM (2007) Tables of integrals, series and functions (7th ed). Aca- demic Press Gupta AK, Chen T (2001) Goodess-of-fit tests for the Skew Normal distribution. Commu- nications in Statistics - Simulation and Computation 30(4):907–930 Loperfido N (2002) Statistical implications of selectively reported inferential results. Statis- tics and Probability Letters 56(1):13–22 Mee RW, Owen D (1983) A simple approximation for bivariate Normal probabilities. Jour- nal of Quality Technology 15(2):72–75 Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb-Douglas production

25 functions with composed error. International Economic Review 18:435–444 Owen D (1956) Tables for computing bivariate Normal probabilities. The Annals of Math- ematical Statistics 27(4):1075–1090 Papadopoulos A (2015) The Half-Normal speciﬁcation for the two-tier stochastic frontier model. Journal of Productivity Analysis 43(2):225–230 Papadopoulos A (2018) The two-tier stochastic frontier framework: Theory and applica- tions, models and tools. PhD Thesis. Athens University of Economics and Business Tong YL (1990) The Multivariate Normal Distribution. Springer-Verlag, New York Tsay WJ, Huang CJ, Fu TT, Ho IL (2013) A simple closed form approximation for the cumulative distribution function of the composite error of stochastic frontier models. Journal of Productivity Analysis 39:259–269 Yalta AT, Schreiber S (2012) Random number generation in gretl. Journal of Statistical Software 50(1):1–13

26 Evaluating the CDF of the Skew-Normal distribution

Christine Amsler Alecos Papadopoulos Peter Schmidt EMEC-D-19-01163 Tables to be included in the main text April 03, 2020

1. Each Table is characterized by a single value of  . Since  1 is the benchmark for assessing performance, Table 1 relates to it, then Table 2 relates to   0.25 , etc.

2. Table 1 consists of four subtables: a) Simulations, b) Approximations, c) Evaluations using

2 Q, 0;  , d) Other evaluations. The Tables for the other values of  consist of three subtables (a,b,c) of which the c-subtables are included in a separate file as Supplementary material.

3. Range of values for = 0.25, 0.50, 1.00, 2.00, 4.00, 8.00

4. Range of values for Q

-28 -24 -20 -16 -12 -8 -6 -4 -2 -1 0 28 24 20 16 12 8 6 4 2 1 32

5. For Q  2 values are given as "1 minus..." exploiting the reflection properties of the Skew Normal CDF.

6. Precision is presented at 5 decimal digits.

7. "DNA" = Does not apply

"***" = value computed as exactly zero or one

"NEG" = value computed as negative.

Page 1 of 13

Table 1,  1 LAMBDA = 1 Table 1a: Simulated evaluations

Exact SN-CDF APS-SIM Φ2 Q AST simulated TONG simulated   simulated (for comparison) -28 6.59973e-345 *** *** *** -24 1.93318e-254 1.93018e-254 1.44902e-314 1.93117e-254 -20 7.58242e-178 7.57298e-178 7.87853e-209 7.57570e-178 -16 4.08161e-115 4.07757e-115 9.89904e-127 4.07757e-115 -12 3.15588e-66 3.15338e-66 4.61302e-68 3.15377e-66 -8 3.87003e-31 3.87003e-31 3.81204e-31 3.86771e-31 -6 9.73356e-19 9.73298e-19 9.75264e-19 9.72809e-19 -4 1.00306e-9 1.00306e-9 1.00335e-9 1.00257e-9 -2 5.17567e-4 5.17567e-4 5.17684e-4 5.17377e-4 -1 2.51714e-2 2.51714e-2 2.51715e-2 2.51649e-2 0 2.50000e-1 2.49935e-1 2.49922e-1 2.49973e-1 1 7.07861e-1 7.07771e-1 7.07720e-1 7.07877e-1 2 1 – 4.49826e-2 1 – 4.50189e-2 1 – 4.50624e-2 1 – 4.48213e-2 4 1 – 6.33423e-5 1 – 6.33965e-5 1 – 6.39875e-5 1 – 5.18460e-5 6 1 – 1.97318e-9 1 – 1.71000e-9 1 – 1.20221e-9 1 – 3.19731e-10 8 1 – 1.24419e-15 1 – 2.58074e-16 1 – 2.65358e-18 1 – 2.07078e-18 12 1 – 3.55298e-33 1 – 3.64006e-39 1 – 2.03340e-52 1 – 1.00259e-44 16 1 – 1.27775e-57 1 – 1.78556e-75 1 – 5.98950e-110 1 – 1.07331e-84 20 1 – 5.50725e-89 1 – 1.55959e-125 1 – 1.41133e-191 1 – 1.85313e-138 24 1 – 2.78078e-127 1 – 1.20169e-189 1 – 5.82277e-296 1 – 4.61373e-206 28 1 – 1.62477e-172 1 – 3.58413e-267 *** 1 – 1.57074e-287 32 1 – 1.09042e-224 *** *** ***

Page 2 of 13

Table 1,  1 LAMBDA = 1 Table 1b: Approximations

Exact SN-CDF APS-LT AZC-LT APS-UT AZC-UT TSAY Φ2 Q lower tail approx. lower tail upper tail approx. upper tail approx. approximation 2 (for comparison) 21QQ     approximation 12  Q 12  QQ -28 6.59973e-345 2.83086e-311 *** -24 1.93318e-254 1.57911e-230 1.93989e-254 -20 7.58242e-178 5.63817e-162 7.62022e-178 -16 4.08161e-115 1.3037e-105 4.11333e-115 -12 3.15588e-66 1.99473e-61 3.19928e-66 DNA DNA -8 3.87003e-31 2.11512e-29 3.98835e-31 -6 9.73356e-19 6.20523e-18 1.02545e-18 -4 1.00306e-9 1.77515e-9 1.11941e-9 -2 5.17567e-4 5.44520e-4 7.28757e-4 -1 2.51714e-2 2.51433e-2 EXACT 5.85499e-2 0 2.50000e-1 2.50166e-1 1 7.07861e-1 7.07833e-1 6.82689e-1 4.83941e-1 2 1 – 4.49826e-2 1 – 4.49557e-2 1 – 4.55003e-2 1 – 1.53910e-2 4 1 – 6.33423e-5 1 – 6.33407e-5 1 – 6.33425e-5 1 – 6.69151e-5 6 1 – 1.97318e-9 1 – 1.97318e-9 1 – 1.97318e-9 1 – 2.02529e-9 8 1 – 1.24419e-15 1 – 1.24419e-15 DNA 1 – 1.24419e-15 1 – 1.26301e-15 12 1 – 3.55298e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.57731e-33 16 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.28270e-57 20 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.52095e-89 24 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78560e-127 28 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62684e-172 32 1 – 1.09042e-224 1 – 1.09042e-224 1 – 1.09042e-224 1 – 1.09148e-224

Page 3 of 13

Table 1,  1 LAMBDA = 1

Table 1c: Evaluations using 22 Q , 0 ; 

Exact SN-CDF Matlab-mvncdf R-mvtnorm R-sn Mathematica Casio-Keisan Φ2 Q Φ2 -MTL Φ2 -R-MVTN Φ2 -R-SN Φ2 -R-MTH Φ2 -CK (for comparison) -28 6.59973e-345 *** NEG *** *** *** -24 1.93318e-254 *** 8.19976e-131 8.19976e-131 *** *** -20 7.58242e-178 *** 3.45994e-93 3.45994e-93 *** *** -16 4.08161e-115 *** NEG *** *** *** -12 3.15588e-66 1.94545e-39 1.94545e-39 1.94545e-39 1.94545e-39 *** -8 3.87003e-31 4.78899e-25 4.78898e-25 4.78898e-25 4.78897e-25 3.87004e-31 -6 9.73356e-19 9.69319e-19 9.69319e-19 9.69319e-19 9.69320e-19 9.73360e-19 -4 1.00306e-9 1.00307e-9 1.00307e-9 1.00307e-9 1.00307e-9 1.00306e-9 -2 5.17567e-4 5.17569e-4 5.17569e-4 5.17569e-4 5.17569e-4 5.17569e-4 -1 2.51714e-2 2.51715e-2 2.51715e-2 2.51715e-2 2.51715e-2 2.51715e-2 0 2.50000e-1 2.50000e-1 2.50000e-1 2.50000e-1 2.50000e-1 2.50000e-1 1 7.07861e-1 7.07861e-1 7.07861e-1 7.07861e-1 7.07861e-1 7.07861e-1 2 1 – 4.49826e-2 1 – 4.49827e-2 1 – 4.49827e-2 1 – 4.49827e-2 1 – 4.49827e-2 1 – 4.49827e-2 4 1 – 6.33423e-5 1 – 6.33415e-5 1 – 6.33415e-5 1 – 6.33415e-5 1 – 6.33415e-5 1 – 6.33415e-5 6 1 – 1.97318e-9 1 – 1.97318e-9 1 – 1.97317e-9 1 – 1.97317e-9 1 – 1.97318e-9 1 – 1.97318e-9 8 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24420e-15 1 – 1.24419e-15 12 1 – 3.55298e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 16 1 – 1.27775e-57 1 – 1.27777e-57 1 – 1.27777e-57 1 – 1.27777e-57 1 – 1.27777e-57 *** 20 1 – 5.50725e-89 1 – 5.50690e-89 1 – 5.50690e-89 1 – 5.50690e-89 1 – 5.50690e-89 *** 24 1 – 2.78078e-127 1 – 2.77996e-127 1 – 2.77996e-127 1 – 2.77996e-127 1 – 2.77996e-127 *** 28 1 – 1.62477e-172 1 – 1.62551e-172 1 – 1.62550e-172 1 – 1.62550e-172 1 – 1.62551e-172 *** 32 1 – 1.09042e-224 1 – 1.09256e-224 1 – 1.09256e-224 1 – 1.09256e-224 1 – 1.09256e-224 ***

Page 4 of 13

Table 1,  1 LAMBDA = 1

Table 1d: Other evaluations

2 Q   QTQ 2 ( , ) 2QQ  2 2  , 0 ;   1 2 2  Q , 0 ;   Mathematica R-sn R-mvtnorm R-mvtnorm Exact SN-CDF Skew Normal cdf OWEN-T 2    12 (for comparison) MATH-SN  2  2 -28 6.59973e-345 6.59973e-345 8.12387e-173 NEG *** -24 1.93318e-254 1.93319e-254 1.39039e-127 8.19976e-131 *** -20 7.58242e-178 7.58245e-178 2.75362e-89 3.45994e-93 *** -16 4.08161e-115 4.08162e-115 6.38875e-58 NEG *** -12 3.15588e-66 3.15589e-66 1.77648e-33 1.94545e-39 *** -8 3.87003e-31 3.87004e-31 1.38361e-6 4.78889e-25 *** -6 9.73356e-19 9.73355e-19 1.24192e-13 9.69312e-19 1.11022e-16 -4 1.00306e-9 1.00307e-9 1.00307e-9 1.00307e-9 1.00307e-9 -2 5.17567e-4 5.17569e-4 5.17569e-4 5.17569e-4 5.17569e-4 -1 2.51714e-2 2.51715e-2 2.51715e-2 2.51715e-2 2.51715e-2 0 2.50000e-1 2.50000e-1 2.50000e-1 2.50000e-1 2.50000e-1 1 7.07861e-1 7.07861e-1 7.07861e-1 7.07861e-1 7.07861e-1 2 1 – 4.49826e-2 1 – 4.49827e-2 1 – 4.49870e-2 1 – 4.49827e-2 1 – 4.49827e-2 4 1 – 6.33423e-5 1 – 6.33415e-5 1 – 6.33415e-5 1 – 6.33415e-5 1 – 6.33415e-5 6 1 – 1.97318e-9 1 – 1.97318e-9 1 – 1.97305e-9 1 – 1.97317e-9 1 – 1.97317e-9 8 1 – 1.24419e-15 1 – 1.24419e-15 *** 1 – 1.24419e-15 1 – 1.33226e-15 12 1 – 3.55298e-33 1 – 3.55296e-33 1 – 1.77648e-33 1 – 3.55296e-33 *** 16 1 – 1.27775e-57 1 – 1.27775e-57 1 – 6.38875e-58 1 – 1.27777e-57 *** 20 1 – 5.50725e-89 1 – 5.50725e-89 1 – 2.75362e-89 1 – 5.50690e-89 *** 24 1 – 2.78078e-127 1 – 2.78078e-127 1 – 1.39039e-127 1 – 2.77996e-127 *** 28 1 – 1.62477e-172 1 – 1.62477e-172 1 – 8.12386e-173 1 – 1.62550e-172 *** 32 1 – 1.09042e-224 1 – 1.09042e-224 1 – 5.45208e-225 1 – 1.09256e-224 ***

Page 5 of 13

Table 2,   0.25 LAMBDA = 0.25 Table 2a: Simulated evaluations and approximations

APS-LT AZC-LT APS-UT AZC-UT APS-SIM TSAY AST simulated lower tail approx. lower tail upper tail approx. upper tail approx. simulated approximation 2 21QQ     approximation 12  Q 12  QQ -28 1.95384e-184 1.95385e-184 3.04471e-183 1.95709e-184 1.99808e-184 -24 2.57692e-136 2.57694e-136 1.54275e-135 2.58211e-136 2.65490e-136 -20 1.48192e-95 1.48199e-95 4.24335e-95 1.48580e-95 1.54506e-95 -16 3.79523e-62 3.79526e-62 6.41224e-62 3.80875e-62 4.03917e-62 -12 4.48929e-36 4.48933e-36 5.44250e-36 4.51401e-36 4.97363e-36 -8 2.63744e-17 2.63747e-17 2.72141e-17 2.66405e-17 3.20914e-17 DNA DNA -6 1.22228e-10 1.22230e-10 1.22451e-10 1.24068e-10 1.64588e-10 -4 9.22945e-6 9.22956e-6 9.18713e-6 9.45846e-6 1.52391e-5 -2 1.26118e-2 1.26120e-2 1.26137e-2 1.32127e-2 3.57804e-2 -1 1.11826e-1 1.11828e-1 1.12071e-1 1.19844e-1 7.04770e-1 0 4.21933e-1 4.22010e-1 4.22779e-1 1 7.94761e-1 7.94501e-1 7.94761e-1 6.82689e-1 4.83941e-2 2 1 – 3.23927e-2 1 – 3.28936e-2 1 – 3.28866e-2 1 – 4.55003e-2 1 – 5.39910e-2 4 1 – 5.41336e-5 1 – 5.41354e-5 1 – 5.41554e-5 1 – 6.33425e-5 1 – 6.69151e-5 6 1 – 1.96740e-9 1 – 1.85219e-9 1 – 1.85072e-9 1 – 1.97318e-9 1 – 2.02529e-9 8 1 – 1.18968e-15 1 – 1.21827e-15 1 – 1.21698e-15 DNA DNA 1 – 1.22219e-15 1 – 1.26307e-15 12 1 – 3.51550e-33 1 – 3.49557e-33 1 – 3.54752e-33 1 – 3.55296e-33 1 – 3.57731e-33 16 1 – 1.15397e-57 1 – 1.11531e-57 1 – 1.27769e-57 1 – 1.27775e-57 1 – 1.28270e-57 20 1 – 3.40590e-89 1 – 3.06768e-89 1 – 5.50724e-89 1 – 5.50725e-89 1 – 5.52095e-89 24 1 – 6.79859e-128 1 – 5.49569e-128 1 – 2.78078e-127 1 – 2.78108e-127 1 – 2.78560e-127 28 1 – 7.46002e-174 1 – 5.29119e-174 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62684e-172 32 1 – 4.00336e-227 1 – 2.46193e-227 1 – 1.09042e-224 1 – 1.09042e-224 1 – 1.09148e-224

Page 6 of 13

Table 2,   0.25 LAMBDA = 0.25

Table 2b: Evaluations using 2 and Owen's T-function

2QQ  2  , 0 ;   22 Q , 0 ;  QTQ 2 ( , )   2   Mathematica Matlab-mvncdf R-mvtnorm R-sn Mathematica R-mvtnorm Skew Normal cdf Φ -MTL Φ -R-MVTN Φ -R-SN Φ -R-MTH 2    2 2 2 2 MATH-SN  2  -28 *** NEG *** *** 1.95512e-184 NEG -24 *** NEG *** *** 2.30646e-136 NEG -20 *** 8.14774e-94 8.14774e-94 8.14774e-94 1.48279e-95 8.14774e-94 -16 4.73218e-62 4.73218e-62 4.73218e-62 4.73219e-62 3.79714e-62 4.73218e-62 -12 4.49135e-36 4.49135e-36 4.49135e-36 4.49135e-36 4.49130e-36 4.49135e-36 -8 2.63840e-17 2.63840e-17 2.63840e-17 2.63840e-17 2.63840e-17 2.63840e-17 -6 1.22266e-10 1.22266e-10 1.22266e-10 1.22266e-10 1.22266e-10 1.22266e-10 -4 9.23165e-6 9.23165e-6 9.23165e-6 9.23165e-6 9.23165e-6 9.23165e-6 -2 1.26138e-2 1.26138e-2 1.26138e-2 1.26138e-2 1.26138e-2 1.26138e-2 -1 1.11839e-1 1.11839e-1 1.11839e-1 1.11839e-1 1.11839e-1 1.11839e-1 0 4.22021e-1 4.22021e-1 4.22021e-1 4.22021e-1 4.22021e-1 4.22021e-1 1 7.94528e-1 7.94528e-1 7.94528e-1 7.94528e-1 7.94528e-1 7.94528e-1 2 1 – 3.28865e-2 1 – 3.28865e-2 1 – 3.28825e-2 1 – 3.28865e-2 1 – 3.28865e-2 1 – 3.28865e-2 4 1 – 5.41108e-5 1 – 5.41108e-5 1 – 5.41108e-5 1 – 5.41108e-5 1 – 5.41108e-5 1 – 5.41108e-5 6 1 – 1.85091e-9 1 – 1.85090e-9 1 – 1.85090e-9 1 – 1.85091e-9 1 – 1.85091e-9 1 – 1.85090e-9 8 1 – 1.21781e-15 1 – 1.21780e-15 1 – 1.21780e-15 1 – 1.21781e-15 1 – 1.21781e-15 1 – 1.21780e-15 12 1 – 3.54847e-33 1 – 3.54847e-33 1 – 3.54847e-33 1 – 3.54847e-33 1 – 3.54847e-33 1 – 3.54847e-33 16 1 – 1.27770e-57 1 – 1.27770e-57 1 – 1.27770e-57 1 – 1.27770e-57 1 – 1.27771e-57 1 – 1.27770e-57 20 1 – 5.50717e-89 1 – 5.50716e-89 1 – 5.50716e-89 1 – 5.50717e-89 1 – 5.50725e-89 1 – 5.50716e-89 24 1 – 2.78138e-127 1 – 2.78137e-127 1 – 2.78137e-127 1 – 2.78138e-127 1 – 2.78078e-127 1 – 2.78137e-127 28 1 – 1.62604e-172 1 – 1.62603e-172 1 – 1.62603e-172 1 – 1.62604e-172 1 – 1.62477e-172 1 – 1.62603e-172 32 1 – 1.09132e-224 1 – 1.09131e-224 1 – 1.09131e-224 1 – 1.09132e-224 1 – 1.09040e-224 1 – 1.09131e-224

Page 7 of 13

Table 3,   0.5 LAMBDA = 0.5 Table 3a: Simulated evaluations and approximations

APS-LT AZC-LT APS-UT AZC-UT APS-SIM TSAY lower tail AST simulated lower tail approx. upper tail approx. upper tail approx. simulated approximation 2 approximatio 21QQ     12  Q 12  QQ n -28 1.01149e-216 1.01153e-216 8.39586e-210 1.01302e-216 1.01943e-216 -24 3.94521e-160 3.94533e-160 2.35208e-155 3.95201e-160 3.98597e-160 -20 3.35008e-112 3.35015e-112 3.69602e-109 3.35715e-112 3.39852e-112 -16 6.34097e-73 6.34108e-73 3.29645e-71 6.35907e-73 6.48057e-73 -12 2.79129e-42 2.79194e-42 1.70533e-41 2.80425e-42 2.89804e-42 -8 3.12590e-20 3.12597e-20 5.36118e-20 3.15241e-20 3.38073e-20 DNA DNA -6 2.10293e-12 2.10299e-12 2.57818e-12 2.13087e-12 2.39355e-12 -4 1.12624e-6 1.12628e-6 1.16936e-6 1.15284e-6 1.44513e-6 -2 5.49841e-3 5.49865e-3 5.47615e-3 5.77508e-3 1.04514e-2 -1 7.24853e-2 7.25140e-2 7.24305e-2 7.83218e-2 2.72607e-1 0 3.52369e-1 3.52389e-1 3.52863e-1 1 7.55168e-1 7.55140e-1 7.55120e-1 6.82689e-1 4.83941e-1 2 1 – 4.00162e-2 1 – 4.00212e-2 1 – 4.00241e-2 1 – 4.55003e-2 1 – 5.39910e-2 4 1 – 6.22789e-5 1 – 6.22206e-5 1 – 6.21731e-5 1 – 6.33425e-5 1 – 6.69151e-5 6 1 – 1.96740e-9 1 – 1.93597e-9 1 – 1.97060e-9 1 – 1.97318e-9 1 – 2.02529e-9 8 1 – 1.18968e-15 1 – 1.07344e-15 1 – 1.24414e-15 DNA DNA 1 – 1.24419e-15 1 – 1.26307e-15 12 1 – 1.32924e-33 1 – 6.73021e-34 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.57731e-33 16 1 – 1.54967e-59 1 – 2.69262e-60 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.28270e-57 20 1 – 6.15395e-94 1 – 3.39492e-95 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.52095e-89 24 1 – 6.42541e-137 1 – 1.08115e-138 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78560e-127 28 1 – 1.58284e-188 1 – 7.98775e-191 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62684e-172 32 1 – 8.75774e-249 1 – 1.31439e-251 1 – 1.09042e-224 1 – 1.09042e-224 1 – 1.09148e-224

Page 8 of 13

Table 3,   0.5 LAMBDA = 0.5

Table 3b: Evaluations using 2 and Owen's T-function

22 Q , 0 ;  QTQ 2 ( , ) 2QQ  2 2  , 0 ;   Mathematica Matlab-mvncdf R-mvtnorm R-sn Mathematica R-mvtnorm Skew Normal cdf Φ -MTL Φ -R-MVTN Φ -R-SN Φ -R-MTH 2    2 2 2 2 MATH-SN  2  -28 *** NEG *** *** 1.01226e-216 NEG -24 *** NEG *** *** 3.94799e-160 NEG -20 *** 2.58637e-95 2.58637e-95 *** 3.35228e-112 2.58637e-95 -16 *** NEG *** *** 7.07475e-73 NEG -12 3.03099e-42 3.03099e-42 3.03099e-42 3.03099e-42 2.56474e-42 3.03094e-42 -8 3.12752e-20 3.12753e-20 3.12753e-20 3.12753e-20 3.12753e-20 3.12753e-20 -6 2.10392e-12 2.10392e-12 2.10392e-12 2.10392e-12 2.10392e-12 2.10392e-12 -4 1.12669e-6 1.12669e-6 1.12669e-6 1.12669e-6 1.12669e-6 1.12669e-6 -2 5.49998e-3 5.49998e-3 5.49998e-3 5.49988e-6 5.49998e-3 5.49998e-3 -1 7.25259e-2 7.25259e-2 7.25259e-2 7.25259e-2 7.25259e-2 7.25259e-2 0 3.52416e-1 3.52416e-1 3.52416e-1 3.52146e-1 3.52416e-1 3.52416e-1 1 7.55215e-1 7.55215e-1 7.55215e-1 7.55215e-1 7.55215e-1 7.55215e-1 2 1 – 4.00003e-2 1 – 4.00003e-2 1 – 4.00003e-2 1 – 4.00003e-2 1 – 4.00003e-2 1 – 4.00003e-2 4 1 – 6.22158e-5 1 – 6.22158e-5 1 – 6.22158e-5 1 – 6.22158e-5 1 – 6.22158e-5 1 – 6.22158e-5 6 1 – 1.97107e-9 1 – 1.97107e-9 1 – 1.97107e-9 1 – 1.97107e-9 1 – 1.97107e-9 1 – 1.97107e-9 8 1 – 1.24416e-15 1 – 1.24416e-15 1 – 1.24416e-15 1 – 1.24416e-15 1 – 1.24416e-15 1 – 1.24416e-15 12 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 16 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 20 1 – 5.50725e-89 1 – 5.50724e-89 1 – 5.50724e-89 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.50724e-89 24 1 – 2.78079e-127 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78079e-127 1 – 2.78078e-127 1 – 2.78078e-127 28 1 – 1.62481e-172 1 – 1.62480e-172 1 – 1.62480e-172 1 – 1.62481e-172 1 – 1.62477e-172 1 – 1.62480e-172 32 1 – 1.09041e-224 1 – 1.09041e-224 1 – 1.09041e-224 1 – 1.09041e-224 1 – 1.09042e-224 1 – 1.09041e-224

Page 9 of 13

Table 4,   2 LAMBDA = 2 Table 4a: Simulated evaluations and approximations

APS-LT APS-UT AZC-UT lower tail approx. AZC-LT APS-SIM TSAY upper tail approx. upper tail approx. AST simulated 2 lower tail simulated approximation 21QQ     approximation 12  Q 12  QQ

-28 *** *** *** *** *** -24 *** *** *** *** *** -20 *** *** *** *** *** -16 1.39078e-282 1.39449e-282 1.63649e-237 1.39328e-282 1.40004e-282 -12 9.89198e-161 9.90990e-161 8.72446e-137 9.88003e-161 9.96493e-161 -8 1.60128e-73 1.60273e-73 5.48345e-64 1.58977e-73 1.62014e-73 DNA DNA -6 7.10964e-43 7.11329e-43 4.78902e-38 7.01062e-43 7.24510e-43 -4 8.12310e-21 8.12496e-21 4.68369e-19 7.88102e-21 8.45184e-21 -2 3.14165e-7 3.14204e-7 5.74526e-7 2.88210e-7 3.61281e-7 -1 1.71791e-3 1.71823e-3 1.83418e-3 1.44377e-3 2.61285e-3 0 1.47519e-1 1.47565e-1 1.47764e-1 1 6.84280e-1 7.05114e-1 6.84524e-1 6.82689e-1 4.83941e-1 2 1 – 4.55569e-2 1 – 2.95521e-2 1 – 4.54997e-2 1 – 4.55003e-2 1 – 5.39910e-2 4 1 – 6.32240e-5 1 – 5.80701e-8 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.69151e-5 6 1 – 4.58049e-10 1 – 2.68437e-21 1 – 1.97318e-9 1 – 1.97318e-9 1 – 2.02529e-9 8 1 – 2.92076e-21 1 – 4.75708e-43 1 – 1.24419e-15 DNA DNA 1 – 1.24419e-15 1 – 1.26307e-15 12 1 – 1.56865e-68 1 – 2.55148e-112 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.57731e-33 16 1 – 3.01540e-153 1 – 3.52533e-216 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.28270e-57 20 1 – 1.23352e-266 *** 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.52095e-89 24 *** *** 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78560e-127 28 *** *** 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62684e-172 32 *** *** 1 – 1.09042e-224 1 – 1.09042e-224 1 – 1.09148e-224

Page 10 of 13

Table 4,   2 LAMBDA = 2

Table 4b: Evaluations using 2 and Owen's T-function

2QQ  2  , 0 ;   22 Q , 0 ;  QTQ 2 ( , )   2   Mathematica Matlab-mvncdf R-mvtnorm R-sn Mathematica R-mvtnorm Skew Normal cdf Φ -MTL Φ -R-MVTN Φ -R-SN Φ -R-MTH 2    2 2 2 2 MATH-SN  2  -28 *** NEG *** *** NEG NEG -24 *** 1.40943e-132 1.40943e-132 *** NEG 1.40943e-132 -20 *** NEG *** *** NEG NEG -16 5.84394e-65 5.84394e-65 5.84394e-65 *** NEG 5.84394e-65 -12 1.94430e-42 1.94430e-42 1.94430e-42 *** NEG 1.94425e-42 -8 2.21374e-28 2.23149e-28 2.23149e-28 3.35266e-30 NEG 2.14373e-28 -6 2.93649e-23 2.93649e-23 2.93649e-23 3.30872e-24 NEG 2.19203e-23 -4 6.77626e-21 5.42101e-20 5.42101e-20 5.42101e-20 NEG NEG -2 3.14362e-7 3.14362e-7 3.14362e-7 3.14362e-7 3.14362e-7 3.14362e-7 -1 1.71888e-3 1.71888e-3 1.71888e-3 1.71888e-3 1.71888e-3 1.71888e-3 0 1.47580e-1 1.47584e-1 1.47584e-1 1.47584e-1 1.47584e-1 1.47584e-1 1 6.84408e-1 6.84408e-1 6.84408e-1 6.84408e-1 6.84408e-1 6.84408e-1 2 1 – 4.54999e-2 1 – 4.54999e-2 1 – 4.54999e-2 1 – 4.54999e-2 1 – 4.54999e-2 1 – 4.54999e-2 4 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33925e-5 1 – 6.33925e-5 1 – 6.33425e-5 6 1 – 1.97318e-9 1 – 1.97317e-9 1 – 1.97317e-9 1 – 1.97318e-9 1 – 1.97318e-9 1 – 1.97317e-9 8 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 12 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 16 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 20 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.50725e-89 24 1 – 2.78077e-127 1 – 2.78077e-127 1 – 2.78077e-127 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78077e-127 28 1 – 1.62478e-172 1 – 1.62478e-172 1 – 1.62478e-172 1 – 1.62478e-172 1 – 1.62477e-172 1 – 1.62478e-172 32 1 – 1.09048e-224 1 – 1.09047e-224 1 – 1.09047e-224 1 – 1.09041e-224 1 – 1.09042e-224 1 – 1.09047e-224

Page 11 of 13

Table 5,   4 LAMBDA = 4 Table 5a: Simulated evaluations and approximations

-28 *** *** *** *** *** -24 *** *** *** *** *** -20 *** *** *** *** *** -16 *** *** *** *** *** -12 *** *** *** *** *** -8 4.01248e-241 4.03754e-241 4.92343e-196 3.99026e-241 4.05474e-241 DNA DNA -6 1.64136e-137 1.64935e-137 1.50631e-113 1.61382e-139 1.65931e-137 -4 2.48724e-63 2.49466e-63 8.82626e-54 2.38047e-63 2.52448d-63 -2 1.91857e-18 1.92038e-18 1.13571e-16 1.66503e-18 2.00571e-18 -1 8.17304e-7 8.17557e-7 1.51799e-6 5.91154e-7 9.52442e-7 0 7.79357e-2 7.79686e-2 7.81201e-2 1 6.82556e-1 6.85512e-1 6.82691e-1 6.82689e-1 4.83941e-1 2 1 – 4.55616e-2 1 – 1.61774e-5 1 – 4.55003e-2 1 – 4.55003e-2 1 – 5.39910e-2 4 1 – 6.32869e-5 1 – 7.87502e-33 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.69151e-5 6 1 – 2.80140e-12 1 – 4.40721e-89 1 – 1.97318e-9 1 – 1.97318e-9 1 – 2.02529e-9 8 1 – 2.65741e-45 1 – 1.04809e-174 1 – 1.24419e-15 DNA DNA 1 – 1.24419e-15 1 – 1.26307e-15 12 1 – 3.8117e-190 *** 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.57731e-33 16 *** *** 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.28270e-57 20 *** *** 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.52095e-89 24 *** *** 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78560e-127 28 *** *** 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62684e-172 32 *** *** 1 – 1.09042e-224 1 – 1.09042e-224 1 – 1.09148e-224

Page 12 of 13

Table 5,   4 LAMBDA = 4

Table 5b: Evaluations using 2 and Owen's T-function

2QQ  2  , 0 ;   22 Q , 0 ;  QTQ 2 ( , )   2   Mathematica Skew Matlab-mvncdf R-mvtnorm R-sn Mathematica R-mvtnorm Normal cdf Φ -MTL Φ -R-MVTN Φ -R-SN Φ -R-MTH 2    2 2 2 2 MATH-SN  2  -28 *** *** *** *** NEG 1.14914e-187 -24 *** *** *** *** NEG NEG -20 *** *** *** *** NEG NEG -16 *** *** *** *** NEG NEG -12 *** *** *** *** NEG NEG -8 4.04318e-241 4.04318e-241 4.04318e-241 *** NEG NEG -6 1.65104e-137 1.65104e-137 1.65104e-137 *** NEG NEG -4 2.49657e-63 2.49657e-63 2.49657e-63 *** NEG NEG -2 1.92153e-18 1.92153e-18 1.92153e-18 1.92154e-18 NEG NEG -1 8.17969e-7 8.17969e-7 8.17969e-7 8.17969e-7 8.17969e-7 8.17969e-7 0 7.79791e-2 7.79791e-2 7.79791e-2 7.79791e-2 7.79791e-2 7.79791e-2 1 6.86290e-1 6.82690e-1 6.82690e-1 6.82690e-1 6.82690e-1 6.82690e-1 2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 4 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 6 1 – 1.97318e-9 1 – 1.97317e-9 1 – 1.97317e-9 1 – 1.97318e-9 1 – 1.97318e-9 1 – 1.97317e-9 8 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 12 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 16 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 20 1 – 5.50725e-89 1 – 5.50724e-89 1 – 5.50724e-89 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.50724e-89 24 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78078e-127 28 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62477e-172 32 1 – 1.09042e-224 1 – 1.09041e-224 1 – 1.09041e-224 1 – 1.09042e-224 1 – 1.09042e-224 1 – 1.09041e-224

Page 13 of 13

Evaluating the CDF of the Skew-Normal distribution

Christine Amsler Alecos Papadopoulos Peter Schmidt Empirical Economics Supplement

Additional Tables and Simulations April 04, 2020 A. Additional Tables

1. Each Table is characterized by a single value of  .

2. Range of values for : 0.25, 0.50, 1.00, 2.00, 4.00, 8.00

3. Since  1 is the benchmark for assessing performance, Table 1 relates to it, then Table 2 relates to   0.25 , etc. Table 1 does not appear here, since the full results for are included in the main paper. What is included here is the c-subtables for   0.25, 0.5, 2, 4 , and the full results for   8.

4. Range of values for Q :

-28 -24 -20 -16 -12 -8 -6 -4 -2 -1 0 28 24 20 16 12 8 6 4 2 1 32

5. For Q  2 values are given as "1 minus..." exploiting the reflection properties of the Skew Normal CDF.

6. Precision is presented at 5 decimal digits.

7. "DNA" = Does not apply

"***" = value computed as exactly zero or one

"NEG" = value computed as negative.

Page 1 of 12

Page 2 of 12

Table 2,   0.25 LAMBDA = 0.25 Table 2c: Other simulations and evaluations

22 Q , 0 ;  QTQ 2 ( , ) 1 2 2  Q , 0 ;   Casio-Keisan R-sn R-mvtnorm TONG simulated Φ2 -CK OWEN-T 122 -28 1.04420e-198 *** 7.80886e-173 *** -24 1.55073e-144 *** 1.17183e-127 *** -20 1.80390e-99 *** 1.60387e-89 *** -16 1.90554e-63 *** 1.70065e-58 *** -12 2.89413e-36 4.49130e-36 1.43112e-34 *** -8 2.61228e-17 2.63840e-17 3.89944e-17 1.11022e-16 -6 1.22228e-10 1.22266e-10 1.22266e-10 1.22266e-10 -4 9.23476e-6 9.23165e-6 9.23165e-6 9.23165e-6 -2 1.26118e-2 1.26138e-2 1.26138e-2 1.26138e-2 -1 1.11833e-1 1.11839e-1 1.11839e-1 1.11839e-1 0 4.21980e-1 4.22021e-1 4.22021e-1 4.22021e-1 1 7.94469e-1 7.94528e-1 7.94528e-1 7.945286e-1 2 1 – 3.29110e-2 1 – 3.28865e-2 1 – 3.28865e-2 1 – 3.28865e-2 4 1 – 5.42607e-5 1 – 5.41108e-5 1 – 5.41108e-5 1 – 5.41108e-5 6 1 – 1.85734e-9 1 – 1.85091e-9 1 – 1.85090e-9 1 – 1.85090e-9 8 1 – 1.56091e-15 1 – 1.21781e-15 1 – 1.20519e-15 1 – 1.22124e-15 12 1 – 8.19260e-34 1 – 3.54847e-33 1 – 3.40985e-33 *** 16 1 – 1.59342e-60 *** 1 – 1.10768e-57 *** 20 1 – 3.15128e-96 *** 1 – 3.90337e-89 *** 24 1 – 4.98777e-141 *** 1 – 1.60895e-127 *** 28 1 – 5.84792e-192 *** 1 – 8.43887e-173 *** 32 1 – 4.90387e-258 *** 1 – 5.48662e-225 ***

Page 3 of 12

Table 3,   0.5 LAMBDA = 0.5 Table 3c: Other simulations and evaluations

22 Q , 0 ;  QTQ 2 ( , ) 1 2 2  Q , 0 ;   Casio-Keisan R-sn R-mvtnorm TONG simulated Φ2 -CK OWEN-T 122 -28 1.07905e-253 *** 8.12387e-173 *** -24 2.46985e-182 *** 1.39039e-127 *** -20 1.74496e-123 *** 2.75335e-89 *** -16 4.30648e-77 1.15928e-70 6.32798e-58 *** -12 5.55937e-43 2.79350e-42 1.43270e-33 *** -8 3.05750e-20 3.12752e-20 3.65487e-18 *** -6 2.10448e-12 2.10392e-12 2.10398e-12 2.10398e-12 -4 1.12718e-6 1.12669e-6 1.12669e-6 1.12669e-6 -2 5.50080e-3 5.49998e-3 5.49998e-3 5.49998e-3 -1 7.25220e-2 7.24853e-2 7.25259e-2 7.25259e-2 0 3.52355e-1 3.52416e-1 3.52416e-1 3.52416e-1 1 7.55117e-1 7.55215e-1 7.55215e-1 7.55215e-1 2 1 – 4.00482e-2 1 – 4.00003e-2 1 – 4.00003e-2 1 – 4.00003e-2 4 1 – 6.25475e-5 1 – 6.22158e-5 1 – 6.22158e-5 1 – 6.22158e-5 6 1 – 1.89849e-9 1 – 1.97311e-9 1 – 1.97107e-9 1 – 1.97107e-9 8 1 – 5.50077e-16 1 – 1.24416e-15 1 – 1.24053e-15 1 – 1.33226e-15 12 1 – 2.80566e-37 1 – 3.55296e-33 1 – 2.12025e-33 *** 16 1 – 9.47987e-71 *** 1 – 6.44953e-58 *** 20 1 – 1.14782e-116 *** 1 – 2.75390e-89 *** 24 1 – 4.30616e-175 *** 1 – 1.39039e-127 *** 28 1 – 4.70839e-246 *** 1 – 8.12386e-173 *** 32 *** *** 1 – 5.45208e-225 ***

Page 4 of 12

Table 4,   2 LAMBDA = 2 Table 4c: Other simulations and evaluations

22 Q , 0 ;  QTQ 2 ( , ) 1 2 2  Q , 0 ;   Casio-Keisan R-sn R-mvtnorm TONG simulated Φ2 -CK OWEN-T 122 -28 *** *** 8.12387e-173 *** -24 *** *** 1.39039e-127 *** -20 *** *** 2.75362e-89 *** -16 1.43125e-318 *** 6.38875e-58 *** -12 1.59676e-165 *** 1.77648e-33 *** -8 1.59013e-73 *** *** *** -6 7.16124e-43 7.11808e-43 *** *** -4 8.13220e-21 8.12094e-21 *** *** -2 3.14404e-7 3.14362e-7 3.14362e-7 3.14362e-7 -1 1.71911e-3 1.71888e-3 1.71888e-3 1.71888e-3 0 1.47510e-1 1.47584e-1 1.47584e-1 1.47584e-1 1 6.84244e-1 6.84408e-1 6.84408e-1 6.84408e-1 2 1 – 4.56067e-2 1 – 4.54999e-2 1 – 4.54999e-2 1 – 4.54999e-2 4 1 – 6.43689e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 6 1 – 1.05533e-10 1 – 1.97318e-9 1 – 1.97317e-9 1 – 1.97317e-9 8 1 – 8.47988e-29 1 – 1.24419e-15 1 – 1.28822e-15 1 – 1.33226e-15 12 1 – 1.99882-e113 1 – 3.55296e-33 1 – 1.77648e-33 *** 16 1 – 1.11031e-263 *** 1 – 6.38875e-58 *** 20 *** *** 1 – 2.75362e-89 *** 24 *** *** 1 – 1.39039e-127 *** 28 *** *** 1 – 8.12386e-173 *** 32 *** *** 1 – 5.45208e-225 ***

Page 5 of 12

Table 5,   4 LAMBDA = 4 Table 5c: Other simulations and evaluations

22 Q , 0 ;  QTQ 2 ( , ) 1 2 2  Q , 0 ;   Casio-Keisan R-sn R-mvtnorm TONG simulated Φ2 -CK OWEN-T 122 -28 *** *** 8.12387e-173 *** -24 *** *** 1.39039e-127 *** -20 *** *** 2.75362e-89 *** -16 *** *** 6.38875e-58 *** -12 *** *** 1.77648e-33 *** -8 3.92791e-241 *** *** *** -6 1.66394e-137 *** *** *** -4 2.49889e-63 *** *** *** -2 1.92145e-18 1.92154e-18 *** *** -1 8.17720e-7 8.17969e-7 8.17969e-7 8.17969e-7 0 7.79354e-2 7.79791e-2 7.79791e-2 7.79791e-2 1 6.82523e-1 6.82690e-1 6.82690e-1 6.82690e-1 2 1 – 4.56071e-2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 4 1 – 6.43245e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 6 1 – 1.15267e-14 1 – 1.97317e-9 1 – 1.97317e-9 1 – 1.97317e-9 8 1 – 6.00173e-71 1 – 1.24419e-16 1 – 1.28822e-15 1 – 1.22124e-15 12 *** 1 – 3.55296e-33 1 – 1.77648e-33 *** 16 *** *** 1 – 6.38875e-58 *** 20 *** *** 1 – 2.75362e-89 *** 24 *** *** 1 – 1.39039e-127 *** 28 *** *** 1 – 8.12386e-173 *** 32 *** *** 1 – 5.45208e-225 ***

Page 6 of 12

Table 6,   8 LAMBDA = 8 Table 6a: Simulated evaluations

APS-SIM AST simulated TONG simulated simulated -28 *** *** *** -24 *** *** *** -20 *** *** *** -16 *** *** *** -12 *** *** *** -8 *** *** *** -6 *** *** *** -4 5.53759e-231 5.61807e-231 5.59389e-231 -2 5.22690e-61 5.263359e-61 5.26260e-61 -1 4.48223e-18 4.49374e-18 4.49634e-18 0 3.95592e-2 3.95631e-2 3.95780e-2 1 6.82565e-1 6.82501e-1 1 – 4.00200e-5 2 1 – 4.55581e-2 1 – 4.55956e-2 1 – 1.69430e-30 4 1 – 6.34124e-5 1 – 6.42924e-5 1 – 1.46418e-164 6 1 – 6.68121e-20 1 – 1.34368e-29 *** 8 1 – 1.59753e-126 1 – 2.75621e-239 *** 12 *** *** *** 16 *** *** *** 20 *** *** *** 24 *** *** *** 28 *** *** *** 32 *** *** ***

Page 7 of 12

Table 6,   8 LAMBDA = 8 Table 6b: Approximations

APS-LT AZC-LT APS-UT AZC-UT TSAY lower tail approx. lower tail upper tail approx. upper tail approx. approximation 2 21QQ     approximation 12  Q 12  QQ -28 *** *** *** -24 *** *** *** -20 *** *** *** -16 *** *** *** -12 *** *** *** DNA DNA -8 *** *** *** -6 *** *** *** -4 6.90427e-186 5.31305e-231 5.61819e-231 -2 1.88167e-51 4.47215e-61 5.32727e-61 -1 2.68045e-16 3.03689e-18 4.70193e-18 0 3.96636e-2 1 6.82689e-1 6.82689e-1 4.83941e-1 2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 5.39910e-2 4 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.69151e-5 6 1 – 1.97318e-9 1 – 1.97318e-9 1 – 2.02529e-9 8 1 – 1.24419e-15 DNA DNA 1 – 1.24419e-15 1 – 1.26307e-15 12 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.57731e-33 16 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.28270e-57 20 1 – 5.50725e-89 1 – 5.50725e-89 1 – 5.52095e-89 24 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78560e-127 28 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.82684e-172 32 1 – 1.09042e-224 1 – 1.09042e-224 1 – 1.09148e-224

Page 8 of 12

Table 6,   8 LAMBDA = 8

Table 6c: Evaluations using 22 Q , 0 ; 

Matlab-mvncdf R-mvtnorm R-sn Mathematica Casio-Keisan

Φ2 -MTL Φ2 -R-MVTN Φ2 -R-SN Φ2 -R-MTH Φ2 -CK -28 *** *** *** *** *** -24 *** *** *** *** *** -20 *** *** *** *** *** -16 *** *** *** *** *** -12 *** *** *** *** *** -8 *** *** *** *** *** -6 *** *** *** *** *** -4 5.60197e-231 5.60197e-231 5.60197e-231 *** *** -2 5.26664e-61 5.26664e-61 5.26664e-61 *** *** -1 4.49905e-18 4.49905e-18 4.49905e-18 4.49898e-18 4.49898e-18 0 3.95834e-2 3.95834e-2 3.95834e-2 3.95834e-2 3.95834e-2 1 6.82689e-1 6.82689e-1 6.82689e-1 6.82689e-1 6.82689e-1 2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 4 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 6 1 – 1.97318e-9 1 – 1.97317e-9 1 – 1.97317e-9 1 – 1.97318e-9 1 – 1.97318e-9 8 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 1 – 1.24419e-15 12 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 1 – 3.55296e-33 16 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 1 – 1.27775e-57 *** 20 1 – 5.50725e-89 1 – 5.50724e-89 1 – 5.50724e-89 1 – 5.50725e-89 *** 24 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78078e-127 1 – 2.78078e-127 *** 28 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62477e-172 1 – 1.62477e-172 *** 32 1 – 1.09042e-224 1 – 1.09041e-224 1 – 1.09041e-224 1 – 1.09042e-224 ***

Page 9 of 12

Table 6,   8 LAMBDA = 8

Table 6d: Other evaluations

QTQ 2 ( , ) 2QQ  2 2  , 0 ;   1 2 2  Q , 0 ;   Mathematica R-sn R-mvtnorm R-mvtnorm Skew Normal cdf OWEN-T 2    12 MATH-SN  2  2 -28 NEG 8.12387e-173 NEG *** -24 NEG 1.39039e-127 NEG *** -20 NEG 2.75362e-89 NEG *** -16 NEG 6.38875e-58 NEG *** -12 NEG 1.77648e-33 NEG *** -8 NEG *** NEG *** -6 NEG *** NEG *** -4 NEG *** NEG *** -2 NEG 3.46945e-18 NEG *** -1 4.49905e-18 *** *** *** 0 3.95834e-2 3.95834e-2 3.95834e-2 3.95834e-2 1 6.82689e-1 6.82689e-1 6.82689e-1 6.82689e-1 2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 1 – 4.55003e-2 4 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 1 – 6.33425e-5 6 1 – 1.97318e-9 1 – 1.97317e-9 1 – 1.97317e-9 1 – 1.97317e-9 8 1 – 1.24419e-15 1 – 1.28822e-15 1 – 1.24419e-15 1 – 1.22124e-15 12 1 – 3.55296e-33 1 – 1.77648e-33 1 – 3.55296e-33 *** 16 1 – 1.27775e-57 1 – 6.38875e-58 1 – 1.27775e-57 *** 20 1 – 5.50725e-89 1 – 2.75362e-89 1 – 5.50724e-89 *** 24 1 – 2.78078e-127 1 – 1.39039e-127 1 – 2.78078e-127 *** 28 1 – 1.62477e-172 1 – 8.12386e-173 1 – 1.62477e-172 *** 32 1 – 1.09042e-224 1 – 5.45208e-225 1 – 1.09041e-224 *** END OF TABLES

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B. Additional Simulations - samples #2 to #5

(see main text, Section 7)

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