Article Type: Research Article Article Citation: Akhilesh Kumar Dubey, Arun Kumar Rao, and Himanshu
Pandey. (2020). BAYESIAN ESTIMATION OF PARAMETERS OF INFLATED GEOMETRIC
DISTRIBUTION. International Journal of Research -GRANTHAALAYAH, 8(5), 126-131. https://doi.org/10.29121/granthaalayah.v8.i5.2020.108 Received Date: 19 May 2020 Accepted Date: 29 May 2020 Keywords: Inflated Geometric
Distribution Joint Beta Prior Precautionary Entropy Loss Functions In this paper Bayes estimators of parameters of inflated geometric distribution have been obtained by taking joint beta prior. The loss functions used are squared, precautionary and entropy.
1. INTRODUCTION
Let us consider the inflated geometric distribution whose probability density function is defined by (1) Where (Singh, Dixit and Roy, 2015). We have
and Thus,
The object of the present paper is to obtain a Bayes estimator of the parameters α and p under different loss functions using a beta prior distribution. A commonly used loss function is the squared error loss function (SELF) (2) The Bayes estimator under the above loss
function is the posterior mean, say,is given by (3) Norstrom
(1996) introduced an alternative asymmetric precautionary loss function and also presented a general class of precautionary loss
functions with quadratic loss function as a special case. A
very useful and simple asymmetric precautionary loss function is given
as . (4) The Bayes estimator under precautionary
loss function is denoted by and is obtained
by solving the following equation. (5) In many practical situations, it appears to be more realistic to express the loss in terms of the ratio . In this case, Calabria and Pulcini (1994) points out that a useful asymmetric loss function is the entropy loss Where and whose
minimum occurs at Also, the loss function has been used
in Dey et al. (1987) and Dey and Liu (1992), in the original form having Thus can written be
as (6) The Bayes estimator under entropy loss
function is denoted by and is obtained
by solving the following equation (7) Since
α and p lies between 0 and 1
so that we may take conjugate prior i.e., Beta distribution and Also α and p are independent then the joint prior is
(8) 2. BAYESIAN ESTIMATION
Let denote a random
sample of size n. Assuming that denotes the
number of observations with value x.
The likelihood function can be expressed as below. (9) Where Using binomial expression, L becomes (10) Under the joint prior as given in (8),
the joint posterior of α and p is given by (11) 2.1. BAYES
ESTIMATORS OF Α
From
(11), the marginal posterior of α
is given by
(12) Thus the Bayes Estimator of
α under squared error loss
function is denoted by and is given by
(13) Using (5), Bayes estimator of α under precautionary loss function
is obtained as
(14) The Bayes estimator of α relative to entropy loss function
using (7), is obtained as (15) 2.2. BAYES
ESTIMATORS OF P
From (11), the marginal posterior of p is given by
(16) The Bayes Estimator of p under squared error loss function is
denoted by and is given by (17) Using (5), Bayes estimator of p under precautionary loss function is
obtained as
(18) The Bayes estimator of p relative to entropy loss function
using (7), is obtained as (19) 3.
CONCLUSION
In this paper, we have obtained a number of Bayes estimator of parameters of inflated
geometric distribution. In equations (13)-(15) we have obtained the Bayes
estimators of α and in equations
(17)-(19), the Bayes estimators of p
under joint beta prior. In the above equation, it is clear
that the Bayes estimators depends upon the parameters of the prior
distribution. In this case the risk functions and corresponding Bayes risks
does not exist. SOURCES OF FUNDINGNone. CONFLICT OF INTERESTNone. ACKNOWLEDGMENTNone. REFERENCES[1] Singh, Dixit and Roy, (2015): Stochastic analysis of infant deaths by age and estimation of parameters. Jour. of Sc. and Tech. 20(1), 1-5. [2] J.G. Norstrom, (1996). The use of precautionary loss functions in risk analysis. IEEE Tran. Rel. Vol. 45, No. 3, 400-403. [3] Calabria, R., and Pulcini, G. (1994): Point estimation under asymmetric loss functions for left truncated exponential samples. Comm. Statist. Theory & Methods, 25 (3), 585-600. [4] D.K. Dey, M. Ghosh and C. Srinivasan (1987): Simultaneous estimation of parameters under entropy loss. Jour. Statist. Plan. And infer., 347-363. [5] D.K. Dey, and Pei-San Liao Liu (1992): On comparison of estimators in a generalized life model. Microelectron. Reliab. 32 (1/2), 207-221.
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