Moment-Ratio Diagrams for Univariate Distributions
ERIK VARGO Department of Mathematics, The College of William & Mary, Williamsburg, VA 23187 RAGHU PASUPATHY The Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA 24061 LAWRENCE LEEMIS Department of Mathematics, The College of William & Mary, Williamsburg, VA 23187
We present two moment-ratio diagrams along with guidance for their interpretation. The first moment- ratio diagram is a graph of skewness versus kurtosis for common univariate probability distributions. The second moment-ratio diagram is a graph of coefficient of variation versus skewness for common univariate probability distributions. Both of these diagrams, to our knowledge, are the most comprehensive to date. The diagrams serve four purposes: (1) they quantify the proximity between various univariate distributions based on their second, third, and fourth moments; (2) they illustrate the versatility of a particular distribution based on the range of values that the various moments can assume; and (3) they can be used to create a short list of potential probability models based on a data set; (4) they clarify the limiting relationships between various well-known distribution families. The use of the moment-ratio diagrams for choosing a distribution that models given data is illustrated.
Key Words: Coefficient of Variation; Kurtosis; Skewness.
Introduction the skewness (or third standardized moment) 3 HE MOMENT-RATIO DIAGRAM for a distribution X − μX T γ3 = E , refers to the locus of a pair of standardized mo- σX ments plotted on a single set of coordinate axes (Kotz and Johnson (2006)). By standardized moments we and the kurtosis (or fourth standardized moment) mean the coefficient of variation (CV), 4 X − μX γ4 = E , σX σX γ2 = , μX where μX and σX are the mean and the standard de- viation of the implied (univariate) random variable X. The classical form of the moment-ratio diagram, plotted upside down, shows the third standardized moment γ (or sometimes its square γ2) plotted as Mr. Vargo is a Ph.D. student at the University of Virginia. 3 3 His email address is [email protected]. abcissa and the fourth standardized moment γ4 plot- ted as ordinate. The plot usually includes all possible Dr. Pasupathy is an Assistant Professor in the Grado De- pairs (γ3,γ4) that a distribution can attain. Because − 2 − ≥ partment of Systems and Industrial Engineering. His email γ4 γ3 1 0 (see Stuart and Ord (1994), Exercise address is [email protected]. 3.19, p. 121), the moment-ratio diagram for a dis- tribution occupies some subset of the shaded region Dr. Leemis is a Professor in the Department of Mathemat- shown in Figure 1. ics. His email address is [email protected].
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moments. Another use for these diagrams has been in getting a sense of the limiting relationships between distributions, and also between various distributions within a system. An excellent example of the latter is the Pearson system of frequency curves where the re- gion occupied by the various distributions comprising 2 the system neatly divides the (γ3 ,γ4) plane (Johnson et al. (1994)).
Since Craig (1936) published the original moment- ratio diagram, various authors have expanded and published updated versions. The most popular of these happen to be the various diagrams appearing in Johnson et al. ((1994), pp. 23, 390). Rodriguez (1977), in clarifying the region occupied by the Burr Type XII distribution in relation to others, provides a fairly comprehensive version of the moment-ratio diagram showing several important regions. Tadika- malla (1980) provides a similar but limited version in clarifying the region occupied by the Burr Type III region.
More recently, Cox and Oakes (1984) have popu- larized a moment-ratio diagram of a different kind— one that plots the CV (γ2) as the abcissa and the third standardized moment (γ3) as the ordinate. Ad- FIGURE 1. The Shaded Region Represents the Set of mittedly, this variation is location and scale depen- Attainable Pairs of Third and Fourth Standardized Moments dent, unlike the classical moment-ratio diagrams in- (γ3, γ4) for Any Distribution. The solid line is the limit γ4 2 volving the third and fourth standardized moments. =1+γ3 for all distributions. Nevertheless, the diagram has become unquestion- ably useful for modelers. A slightly expanded ver- sion of this variation appears in Meeker and Escobar Moment-ratio diagrams, apparently first intro- (1998). duced by Craig (1936) and later popularized by John- son et al. (1994), especially through the plotting of Contribution multiple distributions on the same axes, have found enormous expediency among engineers and statisti- Our contributions in this paper are threefold, cians. The primary usefulness stems from the dia- stated here in order of importance. First, we provide gram’s ability to provide a ready “snapshot” of the a moment-ratio diagram of the CV versus skewness relative versatility of various distributions in terms (γ2,γ3) involving 36 distributions, four of which oc- of representing a range of shapes. Distributions oc- cupy two-dimensional regions within the plot. To our cupying a greater proportion of the moment-ratio knowledge, this is the most comprehensive diagram region are thought to be more versatile owing to available to date. Furthermore, it is the first time the the fact that a larger subset of the allowable mo- entire region occupied by important two-parameter ment pairs can be modeled by the distribution. Ac- families within the CV versus skewness plot (e.g., cordingly, when faced with the problem of having to generalized gamma, beta, Burr Type XII) has been choose a distribution to model given data, model- calculated and depicted. The CV versus skewness ers often estimate the third and fourth standardized plot first appeared in Cox and Oakes (1984) and later moments (along with their standard error estimates) in Meeker and Escobar (1998). The diagrams appear- and plot them on a moment-ratio diagram to get a ing in both these original sources either depict only sense of which distributions may be capable of rep- families with a single shape parameter (e.g., gamma) resenting the shapes implicit in the provided data. or vary only one of the shape parameters while fixing In this sense, a modeler can compare several “candi- all others. Second, we provide a classical moment- date” distributions simultaneously in terms of their ratio diagram (γ3,γ4) that includes 37 distributions,
Journal of Quality Technology Vol. 42, No. 3, July 2010 MOMENT-RATIO DIAGRAMS FOR UNIVARIATE DISTRIBUTIONS 3 four of which occupy two-dimensional regions within the distribution in question, are represented by the plot. While such diagrams are widely available, an unfilled circle (e.g., logistic exponential). the diagram we provide is the most comprehensive (iv) When the boundary of a moment-ratio area is among the sources we know and seems particularly obscured by another area, we include a dotted useful due to its depiction of all distributions in the line (Figure 2) or an arrow (Figure 3) to clarify same plot. In constructing the two moment-ratio di- the location of the obscured boundary. agrams, we have had to derive the limiting behavior (v) When a distribution represented by points in of a number of distributions, some of which seem to one of the moment-ratio diagrams converges be new. Expressions for γ , γ , and γ for some of 2 3 4 as one of its parameters approaches a limiting these distributions are listed in the Appendix. We value (e.g., a t random variable as its degrees of also host the moment-ratio diagrams in a publicly freedom approaches infinity), we often decrease accessible website where particular regions of the di- the font size of the labels to minimize interfer- agram can be magnified for clearer viewing. Third, ence. using an actual data set, we demonstrate what a modeler might do when having to choose candidate (vi) The parameterizations used for the distribu- distributions that “model” given data. tions are from Leemis and McQueston (2008) unless indicated otherwise in the paper. Organization of the Paper The rest of the paper is organized as follows. We The Skewness-Kurtosis Diagram present the two moment-ratio diagrams along with Whether the locus corresponding to a distribu- cues for interpretation in the next section. Follow- tion in Figure 2 is a point, curve, or region usually ing that, we demonstrate the use of the moment- depends on the number of shape parameters. For ex- ratio diagrams for choosing a distribution that mod- ample, the normal distribution has no shape parame- els given data. Finally, we present conclusions and ters and its locus in Figure 2 corresponds to the point suggestions for further research. This is followed by (0, 3). By contrast, because the gamma distribution the Appendix, where we provide analytical expres- has one shape parameter, its locus corresponds to the sions for the moment-ratio locus corresponding to 2 curve γ3 =1.5γ2 + 3. An example of a distribution some of the distributions depicted in the diagrams. that has two shape parameters is the Burr Type XII distribution. It accordingly occupies an entire region Reading the Moment-Ratio Diagrams in Figure 2. In all, Figure 2 has 37 distributions with Two moment-ratio diagrams are presented in this 4 continuous distributions represented by regions, 19 paper. The first, shown in Figure 2, is a plot contain- distributions (15 continuous and 4 discrete) repre- ing the (γ3,γ4) regions for 37 distributions. Figure 3 sented by curves, and 14 distributions (13 continuous is a plot containing the (γ2,γ3) regions for 36 distri- and 1 discrete) represented by one or more points. A butions. For convenience, in both diagrams, we have list of other useful facts relating to Figure 2 follows. chosen to include discrete and continuous distribu- • tions on the same plot. In what follows, we provide The “T” plotted at (γ3,γ4)=(0, 9) corre- a common list of cues that will be useful in reading sponds to the t-distribution with five degrees the diagrams correctly. of freedom, which is the smallest number of de- grees of freedom where the kurtosis exists. (i) Distributions of which moment-ratio regions • The chi-square (S) and Erlang (X) distributions correspond to single points (e.g., normal) are coincide when the chi-square distribution has represented by black solid dots, curves (e.g., an even number of degrees of freedom. This ac- gamma) are represented by solid black lines, counts for the alternating pattern of “S” and and areas (e.g., Burr Type XII) are represented “SX” labels that occur along the curve associ- by colored regions. ated with the gamma distribution. (ii) The names of continuous distributions occupy- • Numerous distributions start at (or include) the ing a region are set in sans serif type; the names locus of the normal distribution and end at (or of continuous distributions occupying a point or include) the locus of the exponential distribu- curve are set in roman type; the names of dis- tion. Two examples of such are the gamma dis- crete distributions occupying a point or curve tribution and the inverted beta distribution. are set in italic type. • Space limitations prevented us from plotting (iii) The end points of curves, when not attained by the values associated with the discrete uniform-
Vol. 42, No. 3, July 2010 www.asq.org 4 ERIK VARGO, RAGHU PASUPATHY, AND LAWRENCE LEEMIS
FIGURE 2. Skewness (γ3) Versus Kurtosis (γ4). (See the supplementary material at http://www.asq.org/pub/jqt/ for a full-color version of this figure.)
Journal of Quality Technology Vol. 42, No. 3, July 2010 MOMENT-RATIO DIAGRAMS FOR UNIVARIATE DISTRIBUTIONS 5
FIGURE 3. CV (γ2) Versus Skewness (γ3). (See the supplementary material at http://www.asq.org/pub/jqt/ for a full-color version of this figure.)
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distribution between its limits as a two-mass Bernoulli curve γ3 = γ2 − 1/γ2 as limits. value with (γ3,γ4)=(0, 1) and its limiting • The regime occupied by the inverted beta dis- distribution (as the number of mass values in- tribution has the gamma curve γ3 =2γ2, creases) with (γ3,γ4)=(0, 1.8). It is plotted as γ2 ∈ (0, 1) and the inverted gamma curve a thick line. − 2 ∈ γ3 =4γ2/(1 γ2 ),γ2 (0, 1) as limits. • The regime occupied by the inverted beta dis- • The regime occupied by the generalized gamma tribution has the curves corresponding to in- distribution has the curves corresponding to the verted gamma and the gamma distributions as power distribution and the Pareto distribution limits. as partial limits. • The regime occupied by the generalized gamma • The regime occupied by the Burr Type XII dis- distribution has the curves corresponding to the tribution has the curves corresponding to the power distribution and the log gamma distribu- Weibull and Pareto distributions as limits. tion as partial limits. • The regime occupied by the Burr Type XII dis- Application tribution has the curve corresponding to the Weibull distribution as a partial limit. The moment-ratio diagrams can be used to iden- • Barring extreme negative skewness values, vir- tify likely candidate distributions for a data set, par- tually all of the regime occupied by the gener- ticularly through a novel use of bootstrapping tech- alized gamma distribution is subsumed by the niques, e.g., Cheng (2006) and Ross (2006). Toward beta distribution. illustrating this, we first formally set up the problem. Let X ,X ,...,X be i.i.d. observations of a random • The beta and the Burr Type XII distributions 1 2 n variable having an unknown cumulative distribution seem complementary in the sense that the beta function (CDF) F (x). Suppose θ is some parame- distribution occupies the “outer” regions of the ter concerning the population distribution (e.g., the diagram while the Burr Type XII distribution coefficient of variation γ ), and let θˆ be its estima- occupies the “inner” regions of the diagram. 2 tor (e.g., the sample coefficient of variationγ ˆ con- Furthermore, the collective regime of the beta 2 structed from X ,X ,...,X ). Also let F (x) denote and Burr Type XII distributions, with a few ex- 1 2 n n the usual empirical CDF constructed from the data ceptions (e.g., Laplace), encompasses all other X ,X ,...,X , i.e., distributions included in the plot. 1 2 n n 1 The CV-Skewness Diagram F (x)= I{X ≤ x}. n n i Unlike in the skewness-kurtosis diagram (Figure i=1 2), the locus of a distribution in the CV-skewness A lot is known about about how well Fn(x) approxi- diagram (Figure 3) depends on the distribution’s lo- mates F (x). For example, the Glivenko–Cantelli the- cation and scale parameters. For this reason, in Fig- orem (Billingsley (1995)) states that Fn → F uni- ure 3, there are fewer distributions (compared with formly in x as n →∞. Furthermore, the deviation of Figure 2) for which the locus is a singleton. Figure 3 Fn(x) from F (x) can be characterized fully through represents a total of 36 distributions with 4 contin- Sanov’s theorem (Dembo and Zeitouni (1998)) under uous distributions represented by regions, 24 distri- certain conditions. butions (19 continuous and 5 discrete) represented We are now ready to demonstrate how the above by curves, and 8 distributions (7 continuous and 1 can be used toward identifying candidate distribu- discrete) represented by one or more points. A list of tions to which a given set of data X ,X ,...,X other useful facts relating to Figure 3 follows. 1 2 n might belong. As usual, the sample mean and sam- • Distributions that are symmetric about the ple standard deviation are calculated as mean have γ3 = 0. Because CV can be adjusted n n to take any value (by controlling the location 1 1 X= X and S = (X − X)2. and scale), symmetric distributions, e.g., error, n i n − 1 i i=1 i=1 normal, uniform, logistic, have the locus γ3 =0 in Figure 3. In order to obtain a nonzero standard deviation, we • The regime occupied by the beta family has assume that at least two of the data values are dis- the gamma curve γ3 =2γ2, γ2 ∈ (0, 1) and the tinct. The point estimates for the CV, skewness, and
Journal of Quality Technology Vol. 42, No. 3, July 2010 MOMENT-RATIO DIAGRAMS FOR UNIVARIATE DISTRIBUTIONS 7 kurtosis are (Figure 3). The first step is to calculate and plot the ∼ S point (ˆγ2, γˆ3) = (0.519, 0.881). We then take B = 200 γˆ2 = , bootstrap samples of n = 23 failure times with re- X 3 placement from the data set. (The value of B was n − 1 Xi X chosen arbitrarily.) The bivariate normal distribution γˆ3 = , n S is fitted to the B data pairs and a concentration el- i=1 4 lipse is then overlaid on the plot of the CV versus n − 1 Xi X skewness as a visual aid to identify likely candidate γˆ4 = , n S distributions for modeling the ball-bearing lifetimes. i=1 The results of this process are displayed in Figure for X= 0. The points (ˆγ3, γˆ4) and (ˆγ2, γˆ3) can be 4, which provides a close-up view of the concentra- plotted in Figures 2 and 3 to give a modeler guidance tion ellipse. In terms of candidate distributions, the concerning which distributions are potential para- following conclusions can be drawn. metric models for statistical inference. Probability distributions in the vicinity of the point estimates • Because ball-bearing lifetimes are inherently are strong candidates for probability models. Unfor- continuous, all of the discrete distributions tunately, these point estimates do not give the mod- should be eliminated from consideration. eler a sense of their precision, so we develop an ap- • The position of the concentration ellipse im- proximate interval estimate in the paragraph below. plies that several distributions associated with Bootstrapping can be used to obtain a measure regions in the (γ2,γ3) graph are candidate dis- tributions: the gamma distribution (and its spe- of the accuracy of the point estimates (ˆγ3, γˆ4) and cial cases), and the Weibull distribution (and (ˆγ2, γˆ3). Let B denote the number of bootstrap sam- ples (a bootstrap sample consists of n observations the Rayleigh distribution as a special case) are drawn with replacement from the original data set). likely to be models that fit the data well. For each bootstrap sample, the two parameters of • The gamma and Weibull distributions both interest (e.g., skewness and kurtosis) are estimated have shape parameters that are greater than using the procedure described in the previous para- 1 within the concentration ellipse, confirming graph and stored. After the B bootstrap samples the intuition that an appropriate model is in have been calculated, the bivariate normal distribu- the increasing failure rate (IFR) class (Cox and tion is fitted to the B data pairs using standard tech- Oakes (1984)) of survival distributions (i.e., the niques. Two of the five parameters of the bivariate ball bearings are wearing out). Consistent with normal distribution, namely, the two sample boot- this conclusion, notice that the point for the strap means, are replaced by the point estimators exponential distribution is far away from the to assure that the bivariate normal distribution is concentration ellipse. centered about the point estimators that were calcu- • Distributions that are close to the concentra- lated and plotted in the previous paragraph. Finally, tion ellipse should also be included as candi- a concentration ellipse is plotted around the point dates. For this data set, the log-normal distri- estimate. The tilt associated with the concentration bution is just outside of the concentration el- ellipse gives the modeler a sense of the correlation lipse, but provides a good fit to the data (see between the two parameters of interest. Crowder et al. (1991), pp. 37–38 and 42–43 for Example details). Any distribution in or near the con- centration ellipse should be considered a can- Consider the n = 23 deep-groove ball-bearing fail- didate distribution. This is confirmed by the 6 ure times (measured in 10 revolutions) four graphs in Figure 5, which show the fitted 17.88 28.92 33.00 41.52 42.12 45.60 Weibull, gamma, log-normal, and exponential 48.48 51.84 51.96 54.12 55.56 67.80 distributions, along with the empirical CDF as- 68.64 68.64 68.88 84.12 93.12 98.64 sociated with the ball-bearing failure data. The 105.12 105.84 127.92 128.04 173.40 three distributions that are inside of or close to the concentration ellipse provide reasonable fits from Lieblein and Zelen (1956), which is discussed in to the data; the exponential distribution, which Caroni (2002). For brevity, we consider the plotting is far away from the concentration ellipse, pro- of the point and associated concentration ellipse for vides a poor fit to the data. only the CV versus skewness moment ratio diagram
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FIGURE 4. Using the CV-Skewness Diagram to Choose a Candidate Distribution (for Modeling Given Data) Through Estimation and Bootstrapping.
The size of the concentration ellipse also gives If point estimates and concentration ellipses are guidelines with respect to sample size. If the con- plotted on both of the moment-ratio diagrams in Fig- centration ellipse is so large that dozens of probabil- ures 2 and 3, the candidate distributions might not ity distributions are viable candidates, then a larger be consistent. The authors believe that the coeffi- sample size is required. As expected, there is gener- cient of variation versus the skewness plot is more ally more variability on the higher-order moments. reliable because it is based on lower order moments. The moment-ratio diagrams can be used in tandem Also, the eccentricity and tilt of the concentration when using any one diagram still leaves a large num- ellipse provide insight on the magnitudes of the vari- ber of candidate distributions. ances of the point estimates and their correlation. For the ball-bearing failure times, the standard error of Conclusions and Further Research the skewness is almost an order of magnitude larger than the standard error of the coefficient of variation. The two moment-ratio diagrams presented in Fig- The slight tilt of the concentration ellipse indicates ures 2 and 3 are useful for insight concerning univari- that there is a small positive correlation between the ate probability distributions and for model discrim- coefficient of variation and the skewness. ination for a particular data set. Plotting a concen-
Journal of Quality Technology Vol. 42, No. 3, July 2010 MOMENT-RATIO DIAGRAMS FOR UNIVARIATE DISTRIBUTIONS 9
FIGURE 5. Empirical and Fitted CDFs for the Weibull, Gamma, Log-Normal, and Exponential Distributions. tration ellipse associated with bootstrap samples on Appendix either chart provides guidance concerning potential In this section, we provide exact expressions for probability distributions that provide an adequate the CV, skewness, and kurtosis for the four distribu- fit to a data set. These diagrams are one of the few tions that occupy (two-dimensional) regions in Fig- ways that data analysts can simultaneously evaluate ures 2 and 3. multiple univariate distributions. Beta Data sets associated with actuarial science, bio- statistics, and reliability engineering often contain The beta family (Johnson et al. (1995), p. 210) censored observations, with right-censored observa- has two shape parameters, p, q > 0, with √ tions being the most common. Plotting the vari- q ous moments is problematic for censored observa- γ2 = ,p,q>0; p2 + pq + p tions. Block and Leemis (2008) provide techniques for overcoming censoring that are based on kernel 2(q − p) 1/p +1/q +1/pq γ3 = ,p,q>0; density function estimation and competing risks the- p + q +2 ory. These techniques can be adapted to produce 2(p + q)2 + pq(p + q − 6) γ =3(p + q +1) , point estimators and concentration ellipses. 4 pq(p + q + 2)(p + q +3) p, q > 0. Further research work associated with these dia- grams would include a Monte Carlo study that eval- The regime in the (γ2,γ3) plane is bounded above uates the effectiveness of the concentration ellipse in by the line γ3 =2γ2 corresponding to the gamma identifying candidate distributions. This study would family, and below by the curve γ3 = γ2 − 1/γ2. The indicate which of the two moment-ratio diagrams is regime in the (γ3,γ4) plane is bounded below by the 2 better for model discrimination. limiting curve γ4 =1+γ3 for all distributions and
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2 above by the curve γ4 =3+(3/2)γ3 corresponding below to the left by the curve corresponding to the to the gamma family. log gamma family. (Recall that the log gamma family with shape parameter α>0 has the rth cumulant Inverted Beta (r) (r) κr =Ψ (α), where Ψ (z) is the (r + 1)th deriva- The beta-prime or the Pearson Type VI fam- tiveoflnΓ(z).) ily (Johnson et al. (1995), p. 248), also known as Burr Type XII the inverted beta family, has two shape parameters, α, β > 0, with The Burr Type XII family (Rodriguez (1977)) has two shape parameters, c, k > 0, with the rth raw α + β − 1 − γ = ,β>2; moment μr =Γ(r/c + 1)Γ(k r/c)/Γ(k),c > 0,k > 2 α(β − 2) 0,r
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Leemis, L. M. and McQueston, J. T. (2008). “Univariate Rodriguez, R. N. (1977). “A Guide to the Burr Type XII Distribution Relationships”. The American Statistician 62, Distributions”. Biometrika 64, pp. 129–134. pp. 45–53. Ross, S. M. (2006). Simulation. Burlington, MA: Elsevier Lieblein, J. and Zelen, M. (1956). “Statistical Investigation Academic Press. of the Fatigue Life of Deep-Groove Ball Bearings”. Journal Stuart, A. and Ord, K. (1994). Kendall’s Advanced Theory of Research of the National Bureau of Standards 57, pp. 273– of Statistics, Volume 1. New York, NY: Hodder Arnold. 316. Tadikamalla, P. R. (1980). “A Look at the Burr and Related Meeker, W. Q. and Escobar, L. A. (1998). Statistical Meth- Distributions”. International Statistical Review 48, pp. 337– ods for Reliability Data. New York, NY: Wiley-Interscience. 344.
∼
Vol. 42, No. 3, July 2010 www.asq.org −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 γ 3 1 Discrete Uniform Arcsin Uniform 2 Gompertz
Standard Muth Benford Triangular 3 C CCCCCCCCCCCCCCCCCCCCC˚CCCCCCCCCCCCC T C T C T SXS T SXS CNSTXT SX T SSX SSX T SSX T SXS S CR T SXS T SXS T SXS Beta T SXS T SXS T SXS T SX T SSX Generalized Gamma T S SXS T SX S T SX T S SX T S C SX 4 T S T SX Logistic S T SX S Hyperbolic 5 T SX Weibull Secant S Gumbel Bernoulli 6 T SX Laplace Exponential Power
7 S Power
Poisson 8 Burr Type XII
Logistic
Inverted Beta Exponential 9 T ESX˚ Chi: C Chi−Square: S ˚ Exponential Erlang: X Power Exponential: E Geometric
10 Log Normal Wald Weibull Normal: N Error Log Logistic Inverted Gamma Lomax, Pareto Gamma γ Rayleigh: R 4 t: T γ 3 Log Inverted Log Exponential Logarithm Logistic Gamma Normal Lomax Wald Weibull Power 4 Pareto Gamma Inverted Beta
3 Generalized S Gamma
Geometric Power
ESX 2 Logistic Poisson ˚ Exponential˚ Burr Type S XII Bernoulli SX Laplace S SX S 1 SX S C SX S SX SXS SXS SXS SXS Benford SXS Beta SXS CR SXS SXSS SXSXS SXSS SXSXS SXSS C SXS SXSXS Error, SXSXSS SXSS SXSSXS SXSSXS C SX SXSXSS SXS SXSS SXS SXS SXSSX S SX S SX C SX S S SX
SXS
S
SX
SXS
S SX C Logistic, CC Muth CC CCC CCCCC Normal, CCCCC CCCCCC CCCCCCC Gompertz CCCCCCCC CCCCC CCCCCCCC C C C Arcsin Uniform, 0 CSX ˚ Discrete Standard Uniform Uniform Chi: C Chi−Square: S Erlang: X Weibull Exponential: E Rayleigh: R −1 Gumbel Exponential Power γ 2 0.0 0.5 1.0 1.5 2.0