Debasis
Bhattacharya
Visva-Bharati
University,India
ABSTRACT
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.