Abstracts
The five parameter gammaWeibull distribution has been introduced by Leipnik and Pearce (2004). Nadarajah and Kotz (2007) have simplified it into four parameter form, using hypergeometric functions in some special cases. We show that the probability distribution function, all moments of positive order and the characteristic function of gammaWeibull distribution of a random variable can be explicitely expressed in terms of the incomplete confluent FoxWright Psifunction, which is recently introduced by Srivastava and Pogány (2007). In the same time, we generalize certain results by Nadarajah and Kotz that follow as special cases of our findings.
gamma distribution; Weibull distribution; confluent FoxWright 1Ψ0; incomplete confluent FoxWright 1Ψ0
A distribuição gammaWeibull a cinco parâmetros foi introduzida for Leipnik e Pearce (2004). Nadarajah e Kotz (2007) a simplificaram para uma forma a quatro parâmetros usando funções hipergeométricas em alguns casos especiais. Nós mostramos que a função de distribuição cumulativa, todos os momentos de ordem positiva e a função característica da distribuição gammaWeibull de uma variável randômica pode ser explicitamente expressa em termos da funçãoPsi de FoxWright confluente incompleta, recentemente introduzida por Srivastava and Pogány (2007). Ao mesmo tempo, generalizamos certos resultados de Nadarajah e Kotz que decorrem como casos especiais de nossos achados.
distribuição gamma; distribuição de Weibull; função 1Ψ0 de FoxWright confluente; função 1Ψ0 de FoxWright confluente incompleta
ENGINEERING SCIENCES
The gammaWeibull distribution revisited
Tibor k. Pogány^{I}; Ram k. Saxena^{II}
^{I}Faculty of Maritime Studies, University of Rijeka, HR51000 Rijeka, Studentska 2, Croatia
^{II}Department of Mathematics and Statistics, Faculty of Science, Jai Narain Vyas University, Jodhpur342004, India
Correspondence to
ABSTRACT
The five parameter gammaWeibull distribution has been introduced by Leipnik and Pearce (2004). Nadarajah and Kotz (2007) have simplified it into four parameter form, using hypergeometric functions in some special cases. We show that the probability distribution function, all moments of positive order and the characteristic function of gammaWeibull distribution of a random variable can be explicitely expressed in terms of the incomplete confluent FoxWright Psifunction, which is recently introduced by Srivastava and Pogány (2007). In the same time, we generalize certain results by Nadarajah and Kotz that follow as special cases of our findings.
Key words: gamma distribution, Weibull distribution, confluent FoxWright _{1}_{Ψ0}, incomplete confluent FoxWright _{1}_{Ψ0}.
RESUMO
A distribuição gammaWeibull a cinco parâmetros foi introduzida for Leipnik e Pearce (2004). Nadarajah e Kotz (2007) a simplificaram para uma forma a quatro parâmetros usando funções hipergeométricas em alguns casos especiais. Nós mostramos que a função de distribuição cumulativa, todos os momentos de ordem positiva e a função característica da distribuição gammaWeibull de uma variável randômica pode ser explicitamente expressa em termos da funçãoPsi de FoxWright confluente incompleta, recentemente introduzida por Srivastava and Pogány (2007). Ao mesmo tempo, generalizamos certos resultados de Nadarajah e Kotz que decorrem como casos especiais de nossos achados.
Palavraschave: distribuição gamma, distribuição de Weibull, função _{1}_{Ψ0} de FoxWright confluente, função _{1}_{Ψ0} de FoxWright confluente incompleta.
1 INTRODUCTION AND RESULTS REQUIRED
Multiplying and then renormalizing the product of densities of gamma and Weibull distributions (Leipnik and Pearce 2004) defined the socalled gammaWeibull distribution. The related probability density function (pdf in the sequel) was simplified by Nadarajah and Kotz (2007, Eq. (1.1)) in the form
where Xa(x) denotes the indicator of {x e A}, while K = K(a, a, r) is the normalizing constant. In what follows we write when the r.v. ξ has the gammaWeibull distribution with parameter θ.
Leipnik and Pearce obtained the characteristic function and the moments of gammaWeibull distribution. Nadarajah and Kotz simplified these results in a modestly simpler form using finite sums of generalized hypergeometric function _{p+1} F_{q} (Nadarajah and Kotz 2007, Eqs. (2.12)), when r = p/q; p, q > 1 are coprime integers.
We will show here that the cumulative distribution function (CDF), the moments of all orders having positive real parts and the characteristic function (chf) can be represented explicitely in terms of the socalled confluent FoxWright function and its incomplete variant form for all r > 0, thereby simplifying and generalizing mainly the results (Nadarajah and Kotz 2007). The incomplete _{p}_{ψ}_{q} is studied very recently by Srivastava and Pogány (2007). In what follows, the symbol _{p}ψ_{q} stands for the FoxWright generalization of the hypergeometric function _{p}F_{q}, with p numerator and q denominator parameters, defined by
for suitably bounded values of x  and in terms of Euler's Gamma function
where an empty product in (2) is to be interpreted (as usual) to be 1 (Mathai and Saxena 1978), (Srivastava et al. 1982). In terms of the analytically continued Gammafunction , the incomplete Gammafunction y (z, a) (Gradshteyn and Ryzhik 2000, 8.350 1.) is given by truncating the integration domain of Gammafunction to [0, ω], i.e.
Following (Srivastava and Pogány 2007, Eq. (6)), we define the socalled incomplete FoxWright function pψq in the form
It is interesting to remark, that the definition (3) immediately yields the limit relationship
2 THE GAMMAWEIBULL gW ( θ) DISTRIBUTION
Cutting off to (0, z) the integration domain _{+} of the integral introduced by Nadarajah and Kotz (2007, Eq. (2.1)), we get the resulting integral
such that it is closely connected to distribution. We express now in terms of the incomplete confluent FoxWright for positive real r ∈ _{+} extending the related result by Nadarajah and Kotz (2007, Eq. (2.2)), which was obtained for rational positive r ∈ .
LEMMA 1.
PROOF. Expanding exp { ax^{r}} into Maclaurin series, interchanging the order of the integration and summation and finally using definition of y(z; ω), then we easily confirm that
which is the assertion of the Lemma. Since the constraint P in (3) being now ensured with r < 1, so the series (7) converges. The case r = 1 results in the incomplete gamma function. Finally, when r > 1, replacing x x^{r} in the integral (5), we easily deduce
This confirms the assertion of the Lemma. □
With the help of (4), we clearly deduce the limiting case
and a fortiori the limiting results of Lemma 1. We, therefore, arrive at
COROLLARY 1.1.
REMARK 1. The case r = 1 in (Nadarajah and Kotz 2007, Eq. (2.2)) is erroneous  there should be Γ(α) in the numerator.
THEOREM 1. Let the random variable , θ > 0 be with pdf (1). Then, the pdf f_{gW}(x) and the PDF F_{gW} (x) of r.v. ξ are of the form:
respectively.
PROOF. To prove (10), we need the exact value of the normalizing constant K, such that appears in (1). But it is straightforward that hance, we deduce that
Expanding the last relation into three formulse with respect to the related domains of r, by the Lemma 1 and Corollary 1.1 we conclude (11).
3 MOMENTS AND CARACTERISTIC FUNCTION
We now derive the moments and the characteristic function of the GammaWeibull distribution. All these results are expressed in terms of the confluent FoxWright function via the integral
THEOREM 2. The moment of order β of the r.v. is given by
whenever
Moreover, the associated is given by
PROOF. Let us take a certain parameter β which satisfies Then, we have
Finally, for the chf, we have
Now, replacing , we immediately obtain (13).
4 CONCLUDING REMARKS
A. The confluent FoxWright functions are defined by series containing gamma function and incomplete gamma function assuming we have
because of (Srivastava et al. 1982, Eq. A25, p. 244)
Finally, applying now the limiting process to (14) and using (3), we deduce
The gamma, incomplete gamma and the confluent hypergeometric are implemented in Mathematica as the subroutines and , respectively.
B. In practice, the parameter vector is unknown, and it has to be estimated from a random sample X = (X_{1}, ... , X_{n}), such that it consists of independently sampled identical random repliacse of the r.v. The traditional parameter estimation procedure is the Maximum Likelihood (ML) method because the ML estimator possesses numerous desired properties, such as asymptotic normality and consistency. Unfortunately, this procedure can be used in a special case only for r = 1, when becomes the ordinary gammadistribution with parameters α and It is known that the related MLestimators are given by
where stands for the familiar digamma function, and denotes the sample mean.
In the case when r ≠ 1, in general the ML estimator does not exist. Indeed, let 0 < r < 1. Then, the loglikelihood function becomes
where
Equating to zero the partial derivatives of l_{0}(X) with respect to μ and a, we conclude
such that it results in
when Now, it is not difficult to find values of 0 for any fixed sample X outside of the manifold (15).
Similar procedure leads to counterexamples when r > 1.
Since the ML method is not efficient when r ≠ 1, following the recent method by Markovic et al. (2009) developed for the threeparameter Weibull distribution, Least Squares parameter estimation method could be derived for the gammaWeibull distribution. However, this problem deserves a separate article.
ACKNOWLEDGMENTS
The authors are thankful to the referees for giving useful suggestions in the improvement of this article. The present investigation was partially supported by the Ministry of Sciences, Education and Sports of Croatia under Research Project No. 11223528182814.
Manuscript received on June 25, 2008; accepted for publication on August 3, 2009
2010 Mathematics subject classification: Primary 33C90; Secondary 62E99.
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Correspondence to:Tibor K. PogányEmail:
Publication Dates

Publication in this collection
11 June 2010 
Date of issue
June 2010
History

Received
25 June 2008 
Accepted
03 Aug 2009