Journal of Statistical Theory and Applications Volume 10, Number 4, 2011, pp. 581-590 ISSN 1538-7887 Characterizations of Weibull Geometric Distribution G. G. Hamedani Department of Mathematics, Statistics and Computer Science Marquette University Milwaukee, WI 53201-1881, USA e-mail: g.hamedani@mu.edu M. Ahsanullah Department of Management Sciences Rider University Lawrenceville, NJ 08648, USA e-mail: ahsan@rider.edu Abstract Various characterizations of the Weibull Geometric distribution are presented. These characterizations are based, on a simple relationship between two truncated moments ; on hazard function ; and on functions of order statistics. Keywords: Exponential geometric distribution ; Rayleigh geometric distribution ; Weibull distribution.
G. G. Hamedani and M. Ahsanullah 582 1 Introduction It is wildly known that the problem of characterizing a distribution is an important problem which has recently attracted the attention of many researchers. Thus, various characterizations have been established in many different directions. The present work deals with the characterizations of a continuous univariate distribution, Weibull Geometric WG) distribution, due to Ortega et al. [15] based, on a simple relationship between two truncated moments ; on hazard function ; and on functions of order statistics. The WG distribution is a special case of Generalized Gamma Geometric GGG) distribution proposed by Ortega et al. [15]. WG is considered to be a suitable distribution for modeling monotone or unimodal failure rates. We refer the reader to the excellent work of Ortega et al. [15] for a detailed discussion as well as applications of WG distribution. The WG distribution depends on scale, shape and, what we call mixing, parameters. GGG depends on an extra shape parameter than WG. Although in many applications an increase in the number of parameters provides a more suitable model, in characterization problems a lower number of parameters without affecting the suitability of the model) is mathematically more appealing see Glänzel and Hamedani [7]), specially in WG case which already has a shape parameter. In view of this observation, our objective here, is to concentrate only on the characterizations of WG distribution. We shall do this in three different directions as discussed in Section 2 below. The pdf probability density function) and cdf cumulative distribution function) of the WG distribution are given, respectively, by f x) = f x; α, β, p) = α β 1 p) βx) α 1 e β x) α 1 pe β x) α) 2, x > 0, 1) and F x) = 1 e β x) α ) 1 pe β x) α) 1, x 0, 2) where α > 0, β > 0 and p 0 < p < 1) are parameters. The parameters α and β are shape and scale and p is mixing parameters, respectively. We would like to mention that Ortega et al. [15] presented seven important GGG distributions, which we believe can be grouped in three groups: {Gamma Geometric and its special case) Chi-Square Geometric}; {Maxwell Geometric and its special case) Half-Normal
Weibull Geometric Distribution 583 Geometric} and {WG and its special cases) Exponential Geometric, Rayleigh Geometric}. We believe the last group is more interesting with a wider domain of applicability and therefore would be the main subject of our work. 2 Characterization Results As we mentioned in the Introduction, the WG distribution and its special cases) may have potential applications in many fields of studies. So, an investigator will be vitally interested to know if their model fits the requirements of the WG distribution. To this end, one will depend on the characterizations of WG distribution which provide conditions under which the underlying distribution is indeed the WG distribution. Throughout this section we assume that the distribution function F is twice differentiable on its support. 2.1 Characterization based on two truncated moments In this subsection we present characterizations of the WG distribution in terms of a simple relationship between two truncated moments. We like to mention here the works of Galambos and Kotz [1], Kotz and Shanbahag [14], Glänzel [2 4], Glänzel et al. [5, 6], Glänzel and Hamedani [7] and Hamedani [8 10] in this direction. Our characterization results presented here will employ an interesting result due to Glänzel [3] Theorem G below). Theorem G. Let Ω, F, P) be a given probability space and let H = [a, b] be an interval for some a < b a =, b = might as well be allowed). Let X : Ω H be a continuous random variable with the distribution function F and let g and h be two real functions defined on H such that E [g X) X x] = E [h X) X x] η x), x H, is defined with some real function η. Assume that g, h C 1 H), η C 2 H) and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation hη = g has no real solution in the interior of H. Then F is uniquely
G. G. Hamedani and M. Ahsanullah 584 determined by the functions g, h and η, particularly x F x) = C η u) η u) h u) g u) exp s u)) du, a where the function s is a solution of the differential equation s = η h η h g and C is a constant, chosen to make H df = 1. Remarks 2.1.1. a) In Theorem G, the interval H need not be closed. b) The goal is to have the function η as simple as possible. For a more detailed discussion on the choice of η, we refer the reader to Glänzel and Hamedani [7] and Hamedani [8 10]. Proposition 2.1.2. Let X : Ω 0, ) be a continuous random variable and let h x) = 1 and g x) = 1 pe β x) α ) 2 for x 0, ). The pdf of X is 1) if and only if the function η defined in Theorem G has the form and η x) = Proof. Let X have pdf 1), then and finally 1 pe β x) α), x > 0. 1 F x)) E [h X) X x] = 1 p)e β x) α 1 pe β x) α) 1, x > 0, 1 F x)) E [g X) X x] = x α β 1 p) βx) α 1 β x) α e du = 1 p) e β x) α, x > 0, η x) h x) g x) = p e β x) α 1 pe β x) α) > 0 for x > 0. Conversely, if η is given as above, then s x) = η x) h x) η x) h x) g x) = α β βx)α 1 1 pe β x) α) 1 = α β βx) α 1 + p α β βx) α 1 e β x) α 1 pe β x) α) 1, x > 0, and hence s x) = βx) α + ln { )} 1 pe β x) α, x > 0. 1 p)
Weibull Geometric Distribution 585 Now, in view of Theorem G, X has cdf 2) and pdf 1). Corollary 2.1.3. Let X : Ω 0, ) be a continuous random variable and let h x) = 1 for x 0, ). The pdf of X is 1) if and only if there exist functions g and η defined in Theorem G satisfying the differential equation Remark 2.1.4. η x) η x) g x) = α ββx)α 1 1 pe β x) α) 1, x > 0. η x) = e βx)α 1 pe β x) α) [ The general solution of the differential equation in Corollary 2.1.3 is g x) α β βx) α 1 e β x) α 1 pe β x) α) 2 dx + D ], for x > 0, where D is a constant. One set of appropriate functions is given in Proposition 2.1.2 with D = 0. Proposition 2.1.5. Let X : Ω 0, ) be a continuous random variable and let h x) = 1 pe β x) α ) 2 and g x) = βx) α 1 pe β x) α ) 2 for x 0, ). The pdf of X is 1) if and only if the function η defined in Theorem G is of the form η x) = 1 + βx) α, x > 0. Proof. Is similar to that of Proposition 2.1.2. 2.2 Characterization based on hazard function For the sake of completeness, we state the following definition. Definition 2.2.1. Let F be an absolutely continuous distribution with the corresponding pdf f. The hazard function corresponding to F is denoted by λ F and is defined by λ F x) = f x) 1 F x), x Supp F, 3) where Supp F is the support of F. It is obvious that the hazard function of a twice differentiable distribution function satisfies the first order differential equation λ F x) λ F x) λ F x) = k x), 4)
G. G. Hamedani and M. Ahsanullah 586 where k x) is an appropriate integrable function. obvious form since f x) f x) = λ F x) λ F x) λ F x), Although this differential equation has an for many univariate continuous distribution 4) seems to be the only differential equation in terms of the hazard function. The goal here is to establish a differential equation which has as simple form as possible and is not of the trivial form 4). For some general families of distributions, however, this may not be possible. distribution. Here is our characterization result for WG Proposition 2.2.2. Let X : Ω 0, ) be a continuous random variable. The pdf of X is 1) if and only if its hazard function λ F satisfies the differential equation λ F x) β α 1) βx) 1 λ F x) = p αβ) 2 βx) 2α 1) e βx)α 1 pe βx)α) 2, x > 0. 5) Proof. If X has pdf 1), then obviously 5) holds. If λ F satisfies 5), then, after some algebra, we can show that d ) βx) α 1) λ F x) dx or λ F x) = = p αβ) 2 βx) α 1 e βx)α 1 pe βx)α) 2, f x) 1 F x) = αβ βx)α 1 1 pe βx)α) 1. Integrating both sides of the above equation with respect to x from 0 to x we obtain ) 1 ln 1 F x)) = βx) α pe βx)α + ln, 1 p from which we arrive at 2). Remark 2.2.3. For characterizations of other well-known continuous distributions based on the hazard function, we refer the reader to Hamedani [11] and Hamedani and Ahsanullah [12]. 2.3 Characterization based on truncated moment of certain functions of order statistics Let X 1:n X 2:n... X n:n be n order statistics from a continuous cdf F. We present here characterization results base on some functions of these order statistics. We refer the reader
Weibull Geometric Distribution 587 to Hamedani et al. [13], among others, for characterizations of other well-known continuous distributions in this direction. Proposition 2.3.1. Let X : Ω 0, ) be a continuous random variable with cdf F such that lim x exp {n 1) βx) α } 1 F x)) n = 0, for some α > 0 and β > 0. E [exp {n 1) βx 1:n ) α } X 1:n > t] = 1 [ p enβt)α 1 1 pe βt) α) n] for some p 0 < p < 1), if and only if X has cdf 2)., t > 0, 6) Then Proof. If X has cdf 2), then clearly 6) is satisfied. Now, if 6) holds, then using integration by parts on the left hand side of lim x exp {n 1) βx) α } 1 F x)) n = 0, we have 6), in view of the assumption n 1) αβ βx) α 1 e n 1)βx)α 1 F x)) n dx t [ 1 = p enβt)α 1 1 pe βt) α) ] n) e n 1)βt)α 1 F t)) n, t > 0. 7) Differentiating both sides of 7) with respect to t, after a lengthy computation, we arrive at f t) 1 F t) = αβ βt)α 1 1 pe βt)α) 1, t > 0. 8) Now, integrating both sides of 8) from 0 to x, we have 1 F x) = 1 p) e βx)α 1 pe βx)α) 1, x 0. Proposition 2.3.2. Let X : Ω 0, ) be a continuous random variable with cdf F. Then )) ] βx E [1 e n:n X n:n < t = 1 p 1 p) n 1 1 pe βx)α) n [1 pe βt)) n ] 1 p) n, t > 0, 9) for some α > 0, β > 0 and p 0 < p < 1), if and only if X has cdf 2). Proof. Is similar to that of Proposition 2.3.1.
G. G. Hamedani and M. Ahsanullah 588 Let X j, j = 1, 2,..., n be n i.i.d. random variables with cdf F and corresponding pdf f and let X 1:n X 2:n... X n:n be their corresponding order statistics. Let X 1:n i+1 be the 1st order statistic from a sample of size n i + 1 of random variables with cdf Then F t x) = F x) F t) 1 F t), x t t is fixed) and corresponding pdf f t x) = fx) 1 F t), x t. X i:n X i 1:n = t) d = X 1:n i+1 d = means equal in distribution), that is f Xi:n X i 1:n x t) = f X 1:n i+1 x) = n i + 1) 1 F t x)) n i f x) 1 F t), x t. Now we can state the following characterization of the WG distribution in yet somewhat different direction. Proposition 2.3.3. Let X : Ω 0, ) be a continuous random variable with cdf F. Then [ { E exp n i)βx i:n ) α} ] X i 1:n = t = 1 p en i+1)βt)α [1 1 pe βt)α) n i+1 ], t > 0, 10) for some α > 0, β > 0 and p 0 < p < 1), if and only if X has cdf 2). Proof. If X has cdf 2), then clearly 10) holds. Now, if 10) holds, then the left hand side of 10) can, in view of the above explanation, be written as { 1 } 1 F t)) n i+1 n i) αβ βx) α 1 e n i)βx)α 1 F x)) n i+1 dx + e n i) βt)α. t The rest of the proof is now similar to that of Proposition 2.3.1. References [1] Galambos, J. and Kotz, S., Characterizations of probability distributions. A unified approach with an emphasis on exponential and related models, Lecture Notes in Mathematics,
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