Bayesian Estimation of the Kumaraswamy Inverse Weibull Distribution

The Kumaraswamy Inverse Weibull distribution has the ability to model failure rates that have unimodal shapes and are quite common in reliability and biological studies. The three-parameter Kumaraswamy Inverse Weibull distribution with decreasing and unimodal failure rate is introduced. We provide a comprehensive treatment of the mathematical properties of the Kumaraswany Inverse Weibull distribution and derive expressions for its moment generating function and the $r$-th generalized moment. Some properties of the model with some graphs of density and hazard function are discussed. We also discuss a Bayesian approach for this distribution and an application was made for a real data set.


Introduction
In this paper we propose a new probability distribution to handle the problem of survival data. Motivated by research developed in recent years, we introduce the Kumaraswany Inverse Weibull distribution that includes several well known distributions used in survival analysis.
Recently, many authors have proposed new classes of distributions, which are modifications of distribution functions which provide hazard ratios contemplating various shapes. We can cite for example the Weibull exponential (Mudholkar et al., 1995), which has also the hazard rate function with a unimodal form, (see also Xie and Lai (1996)). Carrasco et al. (2008) proposed a four-parameter distribution denoted generalized modified Weibull (GMW) distribution, Gusmão et al. (2011) introduced and studied the triparametric inverse Weibull generalized distribution that possesses failure rate with unimodal, increasing and decreasing forms. Pescim et al. (2010) proposed a distribution with four parameters, called beta generalized half normal distribution.
Underexplored in the literature and rarely used by statisticians, the Kumaraswamy distribution (Kumaraswamy, 1980) has a domain in the real interval (0, 1). This property turns the Kumaraswamy distribution a natural candidate to combine with other distributions to produce a more general one. Its cumulative distribution function (cdf) is given by, and its probability density function (pdf) is given by, where a > 0 and b > 0. This density can be unimodal, increasing, decreasing or constant.
Recently, Cordeiro and de Castro (2011) proposed to use the Kumaraswamy to generalize other distributions. Considering that a random variable X has distribution G, they suggest to apply the Kumarasawamy distribution to G(x). Note that, since 0 < G(x) < 1 for any distribution function G, then evaluating equation (1) at G(x) we obtain, where F G is the cdf of the generalized Kumaraswamy-G distribution. Based on these ideas, we consider the Inverse Weibull Distribution as a candidate for G, using Equation (2). Then, performing some adjustments and mathematical manipulations, we obtain the Kumaraswamy inverse Weibull (Kum-IW) distribution.
The rest of the paper is organized as follows. In Section 2, we develop the Kum-IW distribution. Section 3 is devoted to describe basic properties of the distribution. Inference procedures via maximum likelihood and Bayesian approaches are presented in Section 4. Section 5 is devoted to analyze a real data set and in Section 6 we present some conclusions of this work.

Kumaraswamy inverse Weibull distribution
Let T a random variable with inverse Weibull distribution. Then its cdf can be written as, where α > 0, β > 0, and its pdf is given by, Inserting the G function of Equation (3) in Equation (2) it follows that, We note that the parameters a and α in (4) are not identifiable and we adopt the reparameterization c = αa 1/β so that the Kum-IW cdf is rewritten as, where b > 0 and c > 0 are the shape and scale paremeters respectively.
Accordingly, the Kum-IW pdf is now given by, It can be easily seen that when b = 1 we obtain the pdf of the Inverse Weibull (IW) distrbution given by, Finally, the corresponding survival and hazard functions are respectively given by, while the quantile function, Q(u), of the Kum-IW distribution is given by, 1 Some special classes of the Kum-IW The following well known and new distributions are special sub-classes of the Kum-IW distribution.

• Kumaraswamy inverse Rayleigh distribution (Kum-IR)
If β = 2, the Kum-IW distribution reduces to the Kumaraswamy inverse Rayleigh distribution (Kum-IR). Then, with β = 2 the density function of Kum-IW is expressed by: • Inverse Rayleigh distribution (IR) If β = 2 and b = 1, the Kum-IW distribution reduces to the inverse Rayleigh distribution (Kum-IR). Then, with β = 2 and b = 1 the density function of Kum-IW is expressed by: where α > 0, and its pdf is • Kumaraswamy inverse Exponential distribution (Kum-IE) If β = 1, the Kum-IW distribution reduces to the Kumaraswamy inverse Exponential distribution (Kum-IE). Then, with β = 1 the density function of Kum-IW is expressed by: where b > 0 is the shape parameter, and c > 0 is the scale parameter.
Hence, the KIE distribution has two parameters, and its pdf is given by The corresponding survival and hazard functions are respectively • Inverse Exponential distribution (IE) If β = 1 and b = 1, the Kum-IW distribution reduces to the Inverse Exponential distribution (IE). Then, with β = 1 and b = 1 the density function of Kum-IW is expressed by: where λ > 0 and its pdf is

Basic Properties
In this section we describe in detail some properties like expansions, moments, mean deviations, Bonferroni and Lorenz curves, order statistics and entropies which might be useful in any application of the distribution.

Expansions for the distribution and density functions
We now give simple expansions for the cdf of the Kumaraswamy Inverse Weibull distribution. If |x| < 1 and ψ > 0 is a non-integer real number, we If ψ is a positive integer, the series stops at i = ψ. Using expansion in Equation (7) it follows that, and Because the integrals involved in the computation of moments, Bonferroni and Lorenz curves, reliability, Shannon and Rényi entropies and other inferential results do not have analytical solutions, these expansions are necessary.

A general formula for the moments of the Kum-IW
We hardly need to emphasize the need and importance of the moments in any statistical analyses, especially in applied work. Some of the most important features and characteristics of a distribution can be studied using their moments (e.g. tendency, dispersion, skewness and kurtosis). If the random variable T follows the Kum-IW distribution, its k-th moment about zero is given by, The moment generating function M (z) of T for |z| < 1 is, Hence, for |z| < 1, the cumulative generating function of T is We note that it was necessary to use the expansions previously presented for the results of this section.

Mean deviations
The amount of scattering in a population may be measured by all the absolute values of the deviations from the mean or the median. If X is a random variable with Kum-IW distribution with mean µ = E[X] and median M , then the average deviation from the average and the average deviation to the median are defined respectively by, Using the density of extended Kum-IW and given that, where γ(a, x) is the lower incomplete Gamma function, it follows that

Bonferroni and Lorenz curves
Bonferroni and Lorenz curves are widely applied not only in economics to study income and poverty, but also in other fields such as reliability, demography, insurance and medicine.
Let then µ = E(X) e q = F −1 (p; θ), where F −1 (.) is the inverse function of the cumulative function of a random variable X. Bonferroni and Lorenz curves are defined by, Then, using the expanded density, where Γ(a, x) is the upper incomplete gamma function. Therefore, we have, • the Bonferroni curve which is given by: • the Lorenz curve which is given by:

Order statistics and Shannon entropy
Let T 1:n ≤ T 2:n ≤ · · · ≤ T n:n be the order statistics obtained from the Kum-IW distribution. The random variable T r:n , for r = 1, . . . , n, denotes the r-th order statistic in a sample of size n. The pdf of T r:n written as, where f (t) and F (t) are given by equations (6)  The k-th moment µ (k) r:n of the rth order statistic is for k = 1, 2, . . ., and 1 ≤ r ≤ n. Hence, µ (k) r:n = C r:n 1 0 j n−r (1 − j) n−1 dj.
and we obtain an expression for the moment given by, The Shannon entropy of a random variable T is defined as a measure of the quantity of information. A certain message has more quantity of information the greater degree of uncertainty and is defined mathematically is the fdp of T . In particular, for a random variable T which follows the Kum-IW distribution we have, where γ = − ∞ 0 log(j) exp {−j} dj is the approximate value of the Euler's constant.

Rényi entropy
The entropy of a random variable X with density function (8) measuring the uncertainty of the variation. The Rényi entropy is given by, where ρ > 0 and ρ = 1.
In information theory, Rényi entropy generalizes the Shannon entropy.
This form of entropy is important especially in ecology and statistics, where it can be used as an index of diversity. In quantum information, it can be used as a measure of entanglement. If X is a random variable and follows the Kum-IW distribution, then the Rényi entropy is given by

Maximum likelihood estimation
Let T i be a random variable with Kum-IW distribution with parameter vector θ = (b, c, β). The data in survival analysis and reliability studies are generally censored. A very simple random censoring mechanism that is often realistic is one in which each individual i is assumed to have a lifetime T i and a censoring time C i , where T i and C i are independent random variables.
Suppose the data set consists of n independent observations t i = min(T i , C i ) for i = 1, · · · , n. The distribution of C i does not depend on any of the unknown parameters of T i . Parametric inference for such data are usually based on likelihood methods and their asymptotic theory. The censored log-likelihood l(θ) for the model parameters is where F = [1, r] and C = [r + 1, n]; still, C represents the censored data and F represents the failure data. For interval estimation of b, c and β, and tests of hypotheses on these parameters, we must obtain the 3 × 3 observed information matrix J(θ) which is given by, Under conditions met for parameters obeying the parametric space and not considering the limits of the same, the asymptotic distribution of where I(θ) is the expected information matrix. This asymptotic behavior is valid if I(θ) is replaced by J( θ), the observed information matrix evaluated at θ. The asymptotic multivariate normal distribution N 3 (0, J( θ) −1 ) can be used to construct approximate confidence intervals and confidence regions for the individual parameters and for the hazard and survival functions.
The asymptotic normality is also useful for testing goodness of fit of the three parameters the Kum-IW distribution and for comparing this distribution with some of its special submodels using one of the two well-known asymptotically equivalent test statistics -namely, the likelihood ratio (LR) statistic and the Wald and Rao score statistics.

Bayesian approach
Following the Bayesian paradigm, we need to complete the model specification by specifying a prior distribution for the parameters. By Bayes Theorem, the posterior distribution is then proportional to the product of the likelihood function by the prior density.
Subjetivism is the predominant philosophical foundation in Bayesian inference, although in practice noninformative prior densities (built on some formal rule) are frequently used (Kass et al. 1996). Since the parameters in the Kum-IW distribution are all positive quantities and due to the flexibility generated by the two-parameter Gamma distribution this is adopted as prior distribution. So, b ∼ Gamma(m, w), c ∼ Gamma(a, s) and β ∼ Gamma(x, l).
Assuming independence among the prior densities, the posterior density is expressed by, the animals reach the weight of 160kg relative to the period from birth until it weans. We compared the Kaplan-Meyer and Bayesian survival functions through two graphic methods.
Using the expression of ℓ(θ) which a routine was escribed in the software Winbugs see Spiegelhalter et al. (2007) to esteem the values of the vector of parameters θ = (b, c, β) T of the Kum-IW distribution for the data of the cattle of the Nelore race.

Conclusions
We worked a three parameter lifetime distribution called the Kumaraswamy Comparison among survivals generated by the method Bayesian described in the graph S(t) versus Kaplan-Meier.