Asymptotic Minimax bounds for Estimators of
Parameters in a locally Asymptotically Mixture of
Normal Experiments Under Asymmetric Loss

Debasis Bhattacharya
Visva-Bharati University,India

ABSTRACT

There are two broad issues in the asymptotic theory of Statistical Inference
i)the problem of finding the limiting distributions of various statistics to be
used for the purpose of estimation,test of hypotheses,construction of confidence
regions etc. and ii)the problems associated with the questions like how good or
optimal the estimation and testing procedures based on the statistics under
consideration are, how one defines optimality etc.

In a fundamental paper of LeCam (1960) it has been observed that the satisfactory
answers to the above questions will involve the study of the asymptotic
behavior of the likelihood ratio statistic.A very important concept,the concept of
'Limit Experiment' was introduced. In the above context it is an interesting problem
to obtain a lower bound for the risk in estimation in a wide class of competing
estimators and an estimator which attains the bound. A considerable number of works
have been done in this line considering symmetric loss structure, for example
squared error loss. We notice that there are situations where symmertic loss may
not be appropriate to recommend. There are situations where the loss can be different
for equal amount of overestimation and under estimation i.e. there exists a natural
imbalance in the economic results of estimation errors of same magnitude and
of opposite sign. There can be many examples in the field of real estate tax
assesment, agriculture, relibility engineering, pharmaceutics, finance, insurance etc.
As an example, consider a decision problem related to whether the possibly polluted
site needs clean up based on the magnitude of pollutant, duration of exposure etc.
The usual decision rule based on UCL (95%upper confidence limit of the average
concentration) is too conservative. An asymmetric loss function may be very
useful in the above context which will assign different losses for false positive
decision due to over estimation i.e. unnecessary remediation and false negative decision
due to under estimation i.e. no remediation for a polluted site resulting health
hazard.This problem will occur in theory as well, as in the estimation of parameters
in an AR(1) Process and estimating parameter in a Galton Watson Branching
process. These two processes have wide aplications in modelling  different stochastic
events related to finance and insurance where the techniques derived here can be applied.
In this paper we will consider a general class of asymmetric loss function and
obtain a minimax risk bound for estimators of parameters under a LAMN (Locally
Asympotically Mixture of Normal) family of distributions.