Partial Bayes Estimation in a Normal Family

Abstract

Areas of Bayesian analysis became vast throughout several decades. Bayes estimation, empirical
and hierarchical Bayes estimation are important areas among them. In multi-parameter case, notion
of ‘Partial Bayes (PB) Estimation’ is introduced and in N(\theta , \sigma ^2), where parameters being unknown,
‘PB Estimation’ of θ is done by putting the estimator of σ2 obtained by some other methods. When
we do not have enough information regarding the joint parameters of the model of the variable and
when we are estimating one parameter in presence of others, such method may be used instead of
empirical and hierarchical methods when the information about their parameters are not sufficient.
Integrated Risk of the PB estimator and Bayes estimator derived under squared error loss function are
compared along with the classical estimator MLE through simulation technique. From the results, it
is found that PB estimator has almost same risk values as that of the Bayes estimator, whereas the risk
of MLE is higher compared to both of the estimator. For illustrative purpose, a real dataset is used to
apply this PB estimation technique. PB is appropriate name to describe. However, partial Bayes term
has been used in different context of Bayesian hierarchical model and meta analysis.