Abstract

In this study, a Bayesian analysis of Generalized Geometric Series Distribution (GGSD) under different types of loss functions have been studied.

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INTRODUCTION

The probability function of Generalized Geometric Series Distribution (GGSD) was given by Mishra (1982) by using the results of the lattice path analysis as:

(1)

It can be seen that at β = 1, Eq. 1 reduced to simple geometric distribution and is a particular case of Jain and Consul (1971)’s generalized negative binomial distribution in the same way as the geometric distribution is a particular case of the negative binomial distribution.

The various properties and estimation of Eq. 1 have been discussed by Mishra (1982), Mishra and Singh (1992). Hassan et al. (2007) discussed the Bayesian analysis under non-informative and conjugate priors. In this study, the Bayesian analysis of Generalized Geometric Series Distribution (GGSD) under different symmetric loss functions have been studied.

Preliminary theory: Let x be a random variable whose distribution depends on r parameters θ₁, θ₂,...θ_r and let Ω denotes the parameter space of possible values of θ. For the general problem of estimating some specified real-valued function φ(θ) of the unknown parameters θ from the results of a random sample of n observations, we shall assume that φ(θ) is defined for all θ in Ω.

Let x₁, x₂,... x_n be the sample observations. Also, let θ be an estimate of φ(θ) and let be the loss incurred by taking the value of φ(θ) to be . It should be noted that we are restricting consideration hereto loss functions which depend on θ through φ(θ) only. If Ψ(θ) is the prior density of θ then according to Bayes’ theorem the posterior density of θ is where l(θ/x) is the likelihood function of θ given the sample x and:

It follows that for a given x, the expected loss i.e., risk of the estimator is:

(2)

Assuming the existence of Eq. 2 and the sufficient regularity conditions prevail to permit differentiation under the integral sign, the optimum estimator of Φ(θ) will be a solution of the equation:

(3)

The validity of Eq. 3 and the desirability that it should lead to a unique solution necessarily impose restrictions of one’s choice of loss function and prior density of θ. The loss functions which have been considered here are as follow:

 

 

Where δ isa small known quantity:

Where δ₁ and δ₂ are 2 known quantities.

MATERIALS AND METHODS
Bayesian estimation of parameter θ of GGSD under different priors: The likelihood function from Eq. 1 is obtained as:

(4)

Where:

When β is known, the part of the likelihood function which is relevant to Bayesian inference on the unknown parameter θ is .

Bayesian estimation of parameter θ of GGSD under non- informative prior: We assume prior of θ as:

(5)

The posterior distribution of θ from Eq 4. and 5 is:

(6)

The Bayes estimator of parametric function φ(θ) under squared error loss function is the posterior mean which is given as:

If we take φ(θ) = 0, the Bayes estimate of θ is given by:

(7)

This coincides with the moment and ML estimate of θ.

Bayesian estimation of parameter θ of GGSD under beta prior: The more general Bayes estimator of θ can be obtained by assuming the beta distribution as prior information of θ. Thus:

(8)

The posterior distribution of θ is defined as:

(9)

The Bayes estimator of parametric function φ(θ) under squared error loss function is the posterior mean and is given as:

(10)

If we take φ(θ) = 0 then Bayes estimator of θ is given as:

(11)

If a = b = 0, Eq. 11 coincides with Eq. 7. We can consider the more generalized prior as:

(12)

Where c is a positive constant, a and b are known quantities. Under the above loss function, the Bayes’ estimator is given by:

Where:

(13)

and:

(14)

Using Eq. 11, 13 and 14 the Bayes’ estimator under the loss function Eq. 12 is given by:

(15)

 

•	Substituting a = 0 and b = 1, the loss function Eq. 12 becomes the loss function L1 and the Bayes’ estimator under the loss function L1 using Eq. 15 is:

 

 

 

Which is the mean of the posterior distribution:

•	Substituting a = -2 and b = 1, the loss function Eq. 12 becomes the loss function L2 and the Bayes’ estimator under the loss function L2 using Eq. 15 is (Table 1):

 

 

 

Table 1:	Shows the seven different loss functions and the respective Bayes’ estimators of θ under these loss functions
Where δ1 and δ2 are 2 known quantities

 

 

•	Substituting a = 0 and b = 1.2, the loss function Eq. 12 becomes the loss function L3 and the Bayes’ estimator under the loss function L3 using Eq. 15 is:

 

 

 

 

•	Substituting a = 0 and b = 1.2, the loss function Eq. 12 becomes the loss function L4 and the Bayes’ estimator under the loss function L4 using Eq. 15 is:

 

 

 

 

•	The Bayes’ estimator for the zero-one type of loss function L5 under is the mode of the posterior distribution as:

 

 

 

 

•	The Bayes’ estimator for the special zero-one type of loss function L6 is:

 

 

 

RESULTS AND DISCUSSION

As δ₁ = -δ₁ the Bayes’ estimators and are identical. It has been also noted that as β = 1, the above estimators are the Bayesian estimators of the parameter of simple geometric distribution under the above loss functions L₁, L₂, L₃, L₄, L₅, and L₆, respectively.

How to cite this article:

Khurshid Ahmad Mir. On Bayesian Estimation in Generalized Geometric Series Distribution.
DOI: https://doi.org/10.36478/jmmstat.2010.105.108
URL: https://www.makhillpublications.co/view-article/1994-5388/jmmstat.2010.105.108