The gamma – Weibull distribution revisited

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.


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 χ A (x) denotes the indicator of {x ∈ A}, while K = K (α, μ, a, r ) is the normalizing constant.In what follows we write ξ ∼ gW (θ ) 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.

514
TIBOR K. POGÁNY and RAM K. SAXENA 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 Fox-Wright function 1 0 and its incomplete variant form 1 ψ 0 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 q stands for the Fox-Wright generalization of the hypergeometric function p 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 Following (Srivastava and Pogány 2007, Eq. ( 6)), we define the so-called incomplete Fox-Wright function p ψ q in the form It is interesting to remark, that the definition (3) immediately yields the limit relationship Cutting off to (0, z) the integration domain R + of the integral I(α, μ, a, r ) introduced by Nadarajah and Kotz (2007, Eq. (2.1)), we get the resulting integral An Acad Bras Cienc (2010) 82 (2) THE GAMMA-WEIBULL DISTRIBUTION REVISITED 515 such that it is closely connected to gW (θ ) distribution.We express now I gW (θ |ω) in terms of the incomplete confluent Fox-Wright 1 ψ 0 for positive real r ∈ R + extending the related result by Nadarajah and Kotz (2007, Eq. (2.2)), which was obtained for rational positive r ∈ Q + .
PROOF.Expanding exp − ax r into Maclaurin series, interchanging the order of the integration and summation and finally using definition of γ (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.
THEOREM 1.Let the random variable ξ ∼ gW (θ ), 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 K −1 = I gW (θ).As P ξ ≤ 0} = 0 hance, we deduce that Expanding the last relation into three formulae with respect to the related domains of r , by the Lemma 1 and Corollary 1.1 we conclude (11).

MOMENTS AND CHARACTERISTIC FUNCTION
We now derive the moments Eξ β and the characteristic function φ gW (t) of the Gamma-Weibull distribution.All these results are expressed in terms of the confluent Fox-Wright function 1 0 via the integral I gW (θ).

517
THEOREM 2. The moment Eξ β of order β of the r.v.ξ ∼ gW (θ ) is given by Moreover, the associated chf φ gW (t) is given by PROOF.Let us take a certain parameter β, which satisfies α + {β} > 0.Then, we have Finally, for the chf, we have Now, replacing μ − it → μ in I gW (θ ), we immediately obtain ( 13).
B. In practice, the parameter vector θ = (α, μ, a, r ) 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 repliacae of the r.v.ξ ∼ gW (θ ).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 gW (θ ) becomes the ordinary gamma-distribution with parameters α and β := μ + a.It is known that the related ML-estimators are given by ln where ψ = (ln ) = / stands for the familiar digamma function, and X = 1 n n j=1 X j denotes the sample mean.
Since the ML method is not efficient when r = 1, following the recent method by Marković 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.