**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.