Weighted Analogue of Inverse Gamma Distribution : Statistical Properties , Estimation and Simulation Study

In this article we propose a new weighted version of inverse Gamma distribution known as Weighted Inverse Gamma distribution (WIGD). We examine the Length biased and Area biased versions of Weighted Inverse Gamma distribution. Basic structural properties viz moments, mode, moment generating function (mgf), characteristic function (cf), hazard rate function and measures of uncertainty. The parameters of this model are estimated from both classical (namely, maximum likelihood estimator and method of moments, and compare them by using extensive numerical simulations) and Bayesian point of view. The Bayes estimates are estimated by using non-informative Jeffrey’s prior and informative Inverse Chi square prior under different types of loss function (symmetric and asymmetric loss functions). Finally, a simulation study has been conducted for comparing weighted inverse gamma distribution with other competing distributions.


Introduction
The inverse gamma distribution ) , with parameters α and β, is mentioned infrequently in statistical literature, and usually for a specific purpose. One primary use of the IG distribution is for Bayesian estimation of the mean of the one parameter exponential distribution (see for example Johnson et al. (1995), as well as estimating variance in a normal regression. A number of brief descriptions of the properties of the distribution are available, mostly in text books on Bayesian methods, often in the econometrics literature, e.g., Koch (2007) and Poirier (1995). Kleiber and Kotz (2003) list some basic structural properties of the IG distribution and also model incomes with the distribution. Milevsky and Posner (1998) studied the inverse gamma distribution and point out estimation by method of moments.
The learning of weighted distributions can be used for better comprehension of standard distributions, and provides techniques of spreading distributions for further flexibility to fit the data superior. Rao (1965) proposed the concept of weighted distribution, Patil and Rao (1978) discussed how, for example, truncated distributions and damaged observations can give rise to weighted distributions. Weighted distributions occur frequently in research related to bio-medicine, reliability, ecosystem and branching process can be seen in Patil and Rao (1986), Sharma  Lomax distribution with its applications, Das and Roy (2011) discussed the length-biased Weighted Generalized Rayleigh distribution with its properties, also they develop the length-biased Weighted Weibull distribution.
Suppose X is a non-negative random variable with probability density function (pdf) f(x), and then the pdf of the weighted random variable Xw is given by where w(x) be a non-negative weight function. On the support of X, where w(x) > 0 and ω = w(x) f (x) d x is a normalizing constant that forces f w (x) to integrate to 1.
Subject upon the choice of weight function w(x), we will get dissimilar weighted distributions. In this paper, we ruminate w(x) = x c and the model is thus achieved is stated as size biased distribution. Evidently when c = 1, the weight function depends on the length of units of interest, then the resultant distribution is called length biased distribution. Correspondingly, for c = 2, the resultant distribution is called area biased distribution.
This paper is divided in to two parts: first is to study the structural properties of the weighted inverse gamma distribution (WIG) along with its special cases (for c = 1 and 2), and second is to estimate the parameters of the model from both classical and Bayesian view point. Finally, simulation study, summary is provided.

Weighted Inverse Gamma distribution
In this section, we build the pdf of weighted Inverse Gamma distribution by taking the weight function as w(x) = x c and study the behavior of its pdf and hazard function. The probability density function (pdf) and cumulative distribution function (cdf) of the Inverse Gamma distribution is given by denotes the upper incomplete gamma function. , then pdf of X is given by By substituting c = 1 and c = 2 in (4), we get the pdfs of length biased Inverse Gamma (LBIG) and area biased Inverse Gamma (ABIG) distributions respectively. Figure ( The cdf, reliability function and hazard function corresponding to the pdf (4) are respectively given by is the upper incomplete gamma function. and c=0, equation (4) reduces to Levy distribution.

Mode of Weighted Inverse Gamma distribution
is unimodal for given   and c, and achieve its maximum at

Proof:
For the pdf (4), The mode of the weighted inverse gamma can be readily obtained from

Moments of Weighted Inverse Gamma distribution
Moments helps to determine many properties of the distribution such as Averages, dispersion, skewness and kurtosis. The r th moment about origin of the Weighted Inverse Gamma distribution is given by The mean and variance of Weighted Inverse Gamma distribution is given by By substituting c =1, 2 in (5) the mean and variance of Length biased Inverse Gamma and Area biased inverse gamma distribution are By putting c=1,2 in (6) and (7), the skewness and kurtosis of LBIG and ABIG distribution is modified Bessel function of second kind and (.)  is digamma function.

Measures of Uncertainty
The entropy of a random variable X with probability density ) , , (   c WIG is a measure of variation of the uncertainty.

Shannon's Entropy
The Shannon's entropy is given by Using pdf (4) in above equation we get ( )

Methods of Estimation
In this section, parameters of weighted inverse gamma distribution are estimated by various methods of estimation viz, method of moments, maximum likelihood estimation and Bayesian method of estimation.

Method of Moments
In order to estimate the unknown parameters of  On solution the equations (10) and (11) we obtain the estimates for  and  say ˆand ˆ respectively.

Maximum Likelihood Estimation
The MLE ) , ( is obtained by solving the above nonlinear system of equations. It is usually more convenient to use nonlinear optimization algorithms such as quasi-Newton algorithm to numerically maximize the log likelihood function given in (12).
Applying the usual large sample approximation, the MLE   can be treated as being approximately bivariate normal with variance-covariance matrix equal to the inverse of the expected information matrix, i.e.
is the limiting variance-covariance matrix of   .

Bayesian Method of Estimation
In this section, we construct Bayes estimator of the scale parameter  of WIG distribution using non informative Jeffrey's prior and informative Inverse Chi square prior under different loss functions.

Posterior distribution using Jeffrey's prior
Assuming that  has Jeffery prior i.e   1 ) (  g the posterior distribution is given by is a digamma function.

Lemma:
For given posterior distribution (15), we have

Posterior distribution using inverse Chi-square prior
Assuming that  has inverse chi-square defined by

Simulation Study
The most common and simplest method for generating random sample is based on the inverse cumulative distribution function (cdf). For arbitrary cdf, define G −1 (u) = min {x; G(x) ≥ u}. The inverse cdf method can't be directly applied for WIG distribution because of the closed form expression for its quantile function is not available. Here, we intend to use Newton's method for the calculation of the quantile function numerically. The algorithm used for this determination is as follows: