INTRODUCTION
The gamma distribution is used as a life time model (Gupta
and Groll, 1961), though not nearly as much as the Weibull distribution.
It does fit a wide variety of lifetime data adequately, besides failure process
models that lead to it. It also arises in some situations involving the exponential
distribution. Inference for gamma model has been considered by Engelhardt
and Bain (1978), Chao and Glaser (1978), Zaman
et al. (2005), Jamali et al. (2006),
Saat et al. (2008). Lawless
(2003) and Kalbfleisch and Prentice (2002) have made
significant contributions.
The gamma distribution has a pdf of the form:
where, α>0 and k>0 are parameters; α is a scale parameter and k is sometimes called the index or shape parameter.
Also:
This distribution like the Weibull includes the exponential as a special case (k = 1). The distribution with α = 1 is called the one parameter gamma distribution and has pdf:
Bayesian Analysis is an important approach to Statistics, which formally seeks use of prior information and Bayes Theorem provides the formal basis for using this information.
In this approach, parameters are treated as random variables and data is treated
fixed. Ghafoor et al. (2005) and Rahul
et al. (2009) have discussed the applications of Bayesian methods.
ANALYSIS OF GAMMA DISTRIBUTION
Let y = (y_{1}, y_{2},..., y_{n}) be an iid sample from gamma distribution (1) and then likelihood is defined as:
The loglikelihood is given by:
where:
are arithmetic and geometric means, respectively. Setting
and rearranging slightly, we get the likelihood equations:
where:
is a digamma function.
is termed as trigamma function. These functions can be approximated well as:
and
These values are required to implement Newton’s method of optimization.
However, this method is difficult to implement as compared to a very close approximation
discussed by Johnson and Kotz (1970). The maximum estimate
of k can be approximated as:
and
where,
If the value of s ranges from 0 to 0.55722, then value of
is given by Eq. 6 and if it lies between 0.55722 and 17, then
it is given by Eq. 7. Once
is obtained, we can find
from:
For large values of k, we can use the approximation
so that
These estimates are essentially needed for starting iterations.
APPROXIMATION OF GAMMA DISTRIBUTION BASED ON POSTERIOR MODES
In many areas of application, simple models suffice for most practical purposes
but there are occasions when the complexity of the scientific questions at issue
and the data available to answer them warrant the development of more sophisticated
models which depart from standard forms. For such models, approximations to
the posterior distribution of model parameters are useful in their own right
and as a starting point for more exact methods. We make use of Normal and Laplace’s
methods of approximation as discussed by Rubin and Schenker
(1987) and Tierney and Kadane (1986).
Let y_{1}, y_{2},...,y_{n} be an iid observations from a gamma distribution Eq. 1 and then the likelihood is given by:
where,
are arithmetic and geometric mean, respectively.
We define loglikelihood as:
We take partial derivatives with respect to α and k.
We follow the standard approach of Box and Tiao (1973),
Gelman et al. (1995), we assume that a priori α
and k are approximately independent, so that p (α, k)≅ p (α)
p (k) where, p (α) and p (k) are priors for α and k. Using Bayes theorem,
the posterior density p (α, ky) is given by:
The logposterior is given by:
For a prior p (α, k) ≅ p (α) p (k) = 1, we have
The posterior mode is obtained by maximizing Eq. 11 with
respect to α and k. The score vector of log posterior is given by:
and Hessian matrix of log posterior is:
Posterior mode
can be obtained from NewtonRaphson iteration scheme:
Consequently, modal variance Σ can be obtained as:
p (α, ky) can be used for drawing inference about α and k simultaneously.
Using normal approximation, we can write directly a bivariate normal approximation of Eq. 10 as:
Similarly, we can write Bayesian analog of likelihood ratio criterion as:
Using Laplace’s approximation, we can write Eq. 10 as:
The marginal Bayesian inference about α and k is to be based on marginal posterior densities of these parameters. Marginal posterior for α can be obtained after integrating out p (α, k y) with respect to k, i.e.,
Similarly, marginal posterior of k can be obtained as:
We can write normal approximation of marginal posterior p (αy) as:
Bayesian analog of likelihood ratio criterion can also be defined as a test criterion as:
Laplace’s approximation of marginal posterior density p (αy) can be given by:
Similarly, p (ky) can be approximated and results corresponding to normal and Laplace’s approximation can be written as:
or equivalently,
NUMERICAL AND GRAPHICAL ILLUSTRATIONS
The numerical and graphical illustration of posterior densities of the parameters
of interest conveys a very convincing and comprehensive picture of Bayesian
data analysis. We have developed several programs using SPLUS and R softwares
for gamma distribution. These programmes illustrate the strength of Bayesian
methods in various practical situations. Soil samples were collected from rice
growing areas as well as from fruit orchards of Kashmir valley and were analyzed
for some relevant parameters. In present study, we studied available Potassium
in the soil of Kashmir valley. The posterior mode and standard errors of parameters
α and k of gamma distribution are presented in Table 1
by using normal approximation under different types of priors. Graphical display
of marginal of posterior densities for α and k by using Normal approximation
under different priors are shown in Fig. 1af,
whereas Laplace’s approximation for α and k are shown in Fig.
2af.

Fig. 1 (af): 
Normal approximation to parameters alpha and k of gamma distribution
using different priors in SPLUS and R. Postreior density for potassiom
with (a) prior = 1, (b) prior = 1/k, (c) prior = 1/(alpha *k), (d) prior
= 1, (e) prior = 1/k, (f) prior = 1/(alpha *k) 
Table 1: 
Posterior mode and Posterior standard error of Gamma distribution
with different priors 


Fig. 2 (af): 
Laplace’s approximation to parameters alpha and k of
gamma distribution using different priors in SPLUS and R. Postreior density
for potassiom with (a) prior = 1, (b) prior = 1/k, (c) prior = 1/(alpha
*k), (d) prior =1 (e) prior =1/k and (f) prior =1/(alpha *k) 

Fig. 3 (ac): 
Comparing normal and laplace's approximation of alpha of gamma
distribution with different priors using SPLUS and R. Postreior density
for potassiom with (a) prior = 1, (b) prior 1/k and (c) prior = 1/(alpha
*k) 

Fig. 4 (ac): 
Comparing normal and laplace's approximation of k of gamma
distribution with different priors using SPLUS and R. Postreior density
for potassiom with (a) prior = 1, (b) prior 1/k and (c) prior = 1/(alpha
*k) 
The comparison between Normal and Laplace’s methods of approximation for
marginal of posterior densities for parameters α and k are shown in Fig.
3ac and Fig. 4ac,
respectively. This graphical comparison shows that the two approximations are
in close agreement.