PROOF OF CONJECTURES ON THE STANDARD DEVIATION, SKEWNESS AND KURTOSIS OF THE SHIFTED GOMPERTZ DISTRIBUTION Authors: Fernando Jimenez´ Torres { Dpto. de M´etodos Estad´ısticos,Universidad de Zaragoza, Mar´ıade Luna 3, Campus R´ıoEbro, 50018, Zaragoza, Spain [email protected] Abstract: • Three conjectures on the standard deviation, skewness and kurtosis of the shifted Gompertz distribution, as the shape parameter increases to +1, are proved. In this regard, the exponential integral function and polygamma functions are used in the proofs. In addition, an explicit expression for the ith moment of this probabilistic model is obtained. These results allow to place the shifted Gompertz distribution in the Skewness{Kurtosis diagram, providing a valuable help in the decision to choose the shifted Gompertz distribution among the models to fit data. Their usefulness is illustrated by fitting a real malaria data set using the maximum likelihood method for estimating the parameters of the shifted Gompertz distribution and some classical models. Goodness-of-fit measures are used to compare their performance. Key-Words: • Shifted Gompertz distribution; Moment generating function; Skewness{Kurtosis dia- gram; Polygamma functions. AMS Subject Classification: • 60E05, 62E15, 26B20. 2 F. Jim´enezTorres Proof of conjectures of the shifted Gompertz distribution 3 1. INTRODUCTION One of the most important references in models of adopting timing of inno- vations is the model of Bass [1]. From this model, Bemmaor [3] formulated that the individual-level model of adopting timing of a new product in a market is ran- domly distributed according to the shifted Gompertz distribution. More recently, Lover et al. [6] show that modeling studies of period of time to first relapse in human infections with malaria in the New World tropical region, can support the shifted Gompertz distribution. Some statistical properties of the shifted Gompertz distribution were ob- tained in Bemmaor [3]. Jim´enezTorres and Jodr´a[8] gave explicit expressions for the first and second moment, a closed form expression for the quantile function was derived, and the limit distributions of extreme order statistics were conside- red. In Jim´enezTorres [7] the method of least squares, method of maximum likelihood and method of moments to estimate the parameters of the shifted Gompertz distribution were used. In this paper we want to expand and complete the knowledge and statistical properties of the shifted Gompertz distribution, solving the three conjectures presented in Jim´enezTorres and Jodr´a[8] and ob- taining a general expression for the moments. Although the Gompertz distribution Z has been given in different forms in the literature, the cumulative distribution function (cdf) FZ (z) = P (Z ≤ −αe−βz z) = e , −∞ < z < +1, found in Bemmaor [3], satisfiesp thatp its standard deviation, skewness and excess kurtosis are equals to π=( 6β), 12 6ζ(3)/π3 and 2:4, respectively, where ζ(·) denotes the Riemann zeta function. The skewness of 3 3 a random variable X is defined by γ1 = E[(X − µ) ]/σ and is a measure of the asymmetry of the probability distribution. The excess kurtosis of X is given by 4 4 γ2 = E[(X − µ) ]/σ − 3 and it describes the shape of the tails of the probability distribution. Let X be a random variable having the shifted Gompertz distribution with parameters α and β, where α > 0 is a shape parameter and β > 0 is a scale parameter. The probability density function of X is ( ) −(βx+αe−βx) −βx (1.1) fX (x) = βe (1 + α 1 − e ) x > 0: This model can be characterized as the maximum of two independent ran- dom variables with Gompertz distribution (parameters α > 0 and β > 0) and ex- ponential distribution (parameter β > 0). From (1.1), given that limα!0 fX (x) = βe−βx, it may be noted that the shifted Gompertz distribution gets close to an exponential distribution with mean 1/β, as the parameter α decreases to 0. So, for a fixed value of β, limα!0 σ = 1/β, where σ is the standard deviation of X. For the shifted Gompertz distribution we have limα!0 γ1 = 2 and limα!0 γ2 = 6, 4 F. Jim´enezTorres which are the skewness and kurtosis of the exponential distribution. If the shape parameter α increases to infinity, the asymptotic behavior of the shifted Gom- pertz distribution is nontrivial and these limits require analytic tools for their calculation. Based on numerical evidence showed in Jim´enezTorres and Jodr´a[8] the next three conjectures were presented: π Conjecture 1 : lim σ = p α!+1 6β p 12 6ζ(3) Conjecture 2 : lim γ1 = α!+1 π3 Conjecture 3 : lim γ2 = 2:4 α!+1 The remainder of this note is organized as follows. In Section 2, we prove Conjecture 1. In Section 3, we provide an explicit expression for the ith moment of the shifted Gompertz distribution. In Section 4 and Section 5, we prove Con- jecture 2 and Conjecture 3, respectively. In Section 6 we show the importance of these results in the choice of the shifted Gompertz distribution among the models to fit a real data set and finally, the main conclusions are presented in Section 7. 2. PROOF OF CONJECTURE 1 In Jim´enezTorres and Jodr´a[8] explicit expressions for the moments of orders 1 and 2 of X were obtained. The first moment of X, or mean µ of X, is ( ) 1 1 − e−α (2.1) E[X] = γ + log(α) + E (α) + ; β 1 α ≈ where γ 0:57721 is the Euler{MascheroniR constant and E1(x) is the exponential +1 e−t integral function, defined by E1(x) = x t dt, x > 0. The second moment of X is ( ) 2 (2.2) E[X2] = γ + log(α) + E (α) + F [1; 1; 1; 2; 2; 2; −α]α2 ; αβ2 1 3 3 P − − +1 (−α)k 1 where 3F3[1; 1; 1; 2; 2; 2; α] = k=1 k!k2 is a generalized hypergeometric func- tion. Moreover, we need the next expression (see Geller and Ng [5]) for a > 0 and b > 0 Z 1 +1 E (ax) 1( ) X+ (−ab)k (2.3) 1 dx = (γ + log(ab))2 + ζ(2) + ; x 2 k!k2 b k=1 Proof of conjectures of the shifted Gompertz distribution 5 π2 where ζ(2) = . In particular, using (2.3) with a = 1 and b = α, we obtain 6 Z 1 +1 E (x) 1( ) X+ (−α)k (2.4) 1 dx = (γ + log(α))2 + ζ(2) + ; x 2 k!k2 α k=1 and in the next theorem, we prove Conjecture 1. Theorem 2.1. The limit of the standard deviation, σ, of the shifted Gom- pertz distribution X as the shape parameter α increases to +1 is finite and its value is π (2.5) lim σ = p : α!+1 6β Proof: The variance of a random variable X is σ2 = E[X2] − (E[X])2. From (2.1), (2.2) and (2.4) we have [ Z ] 2 +1 E (x) α( ) σ2 = γ + log(α) + E (α) − α 1 dx + (γ + log(α))2 + ζ(2) αβ2 1 x 2 ( α ) 1 1 − e−α 2 (2.6) − γ + log(α) + E (α) + β2 1 α ζ(2) = + R(α); β2 where Z 2 ( ) 2 +1 E (x) (2.7) R(α) = γ + log(α) + E (α) − 1 dx αβ2 1 β2 x ( α ) 1 ( ) 1 1 − e−α 2 + (γ + log(α) 2 − γ + log(α) + E (α) + : β2 β2 1 α So, lim σ2 = ζ(2)/β2 + lim R(α). Now, in (2.7) we take limit as α increases α!+1 α!+1 to +1, taking into account the next limits related to the exponential integral function (see Geller and Ng [5]): ( ) ( ) ( ) −x p (2.8) lim log(x)E1(x) = lim e E1(x) = lim x E1(x) = 0: x!+1 x!+1 x!+1 So, lim R(α) = 0, and Conjecture 1 is proved. α!+1 To prove Conjecture 2 and Conjecture 3 we need expressions of the moments of orders 3 and 4, respectively. In Section 3 we are more ambitious and obtain a general expression for the moment of order i of the shifted Gompertz distribution. 6 F. Jim´enezTorres 3. MOMENT OF ORDER i OF X R i +1 i The ith moment of X, denoted and defined by E[X ] = 0 x fX (x)dx, i = 1; 2;:::, where fX (x) is given in (1.1), does not seem to have a closed-form expression in terms of elementary functions, but we can find a series expansion. Let γ(a; b) be the lower incomplete gamma function defined for any a > 0 and b > 0 by Z b (3.1) γ(a; b) = va−1e−vdv; 0 [ ] tX and let MX (t) be the moment generating function of X, i.e., MX (t) = E e . In the next theorem we obtain an expression of this function. Theorem 3.1. The moment generating function of the shifted Gompertz distribution X for jtj < β is t/β−1 −α (3.2) MX (t) = α (α + t/β)γ(1 − t/β; α) + e : Proof: By definition, we have Z Z [ ] +1 +1 tX tx tx−βx−αe−βx −βx E e = e fX (x)dx = β e (1 + α(1 − e ))dx 0 Z 0 Z +1 +1 −βx −βx (3.3) = (1 + α)β etx−βx−αe dx − αβ etx−2βx−αe dx: 0 0 The change of variable v = αe−βx in (3.3) provides Z Z [ ] α α E etX = αt/β−1(1 + α) e−vv−t/βdv − αt/β−1 e−vv1−t/βdv ( 0 0 ) (3.4) = αt/β−1 (1 + α)γ(1 − t/β; α) − γ(2 − t/β; α) : Integrating by parts in (3.1) yields the recurrence relation γ(a+1; b) = aγ(a; b)− bae−b.