The gamma-Weibull distribution revisited

ENGINEERING SCIENCES

The gamma-Weibull distribution revisited

Tibor k. Pogány^I; Ram k. Saxena^II

^IFaculty of Maritime Studies, University of Rijeka, HR-51000 Rijeka, Studentska 2, Croatia

^IIDepartment of Mathematics and Statistics, Faculty of Science, Jai Narain Vyas University, Jodhpur-342004, India

^{Correspondence to}

ABSTRACT

The five parameter gamma-Weibull 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 gamma-Weibull distribution of a random variable can be explicitely expressed in terms of the incomplete confluent Fox-Wright Psi-function, 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 Fox-Wright _1Ψ0, incomplete confluent Fox-Wright _1Ψ0.

RESUMO

A distribuição gamma-Weibull 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 gamma-Weibull de uma variável randômica pode ser explicitamente expressa em termos da função-Psi de Fox-Wright 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.

Palavras-chave: distribuição gamma, distribuição de Weibull, função _1Ψ0 de Fox-Wright confluente, função _1Ψ0 de Fox-Wright 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 so-called gamma-Weibull 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 gamma-Weibull distribution with parameter θ.

Leipnik and Pearce obtained the characteristic function and the moments of gamma-Weibull 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. (2.1-2)), when r = p/q; p, q > 1 are co-prime 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 so-called confluent Fox—Wright function and its incomplete variant form for all r > 0, thereby simplifying and generalizing mainly the results (Nadarajah and Kotz 2007). The incomplete _pψq is studied very recently by Srivastava and Pogány (2007). In what follows, the symbol _pψ_q stands for the Fox-Wright generalization of the hypergeometric function _pF_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 Gamma-function , the incomplete Gamma—function y (z, a) (Gradshteyn and Ryzhik 2000, 8.350 1.) is given by truncating the integration domain of Gamma-function to [0, ω], i.e.

Following (Srivastava and Pogány 2007, Eq. (6)), we define the so-called incomplete Fox-Wright function pψq in the form

It is interesting to remark, that the definition (3) immediately yields the limit relationship

2 THE GAMMA-WEIBULL 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 Fox—Wright 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 Gamma-Weibull distribution. All these results are expressed in terms of the confluent Fox-Wright 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 Fox-Wright 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 = (X₁, ... , 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 gamma-distribution with parameters α and It is known that the related ML-estimators 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 log-likelihood function becomes

where

Equating to zero the partial derivatives of l₀(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 three-parameter Weibull distribution, Least Squares parameter estimation method could be derived for the gamma-Weibull 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. 112-2352818-2814.

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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Publication Dates

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

History

• Received
  25 June 2008
• Accepted
  03 Aug 2009