ABSTRACT
In Chapter 3, we have discussed how to estimate a parameter θ given a set of observations X = (X1,X2, . . . ,Xn)′ following a joint probability distribution, f(x| θ), θ ∈ Θ. (For the sake of simplicity we assume that the distribution of X depends only on a single, scalar valued parameter θ. Multiparameter cases, i.e., vector valued parameter θ can be considered suitably if the situation demands.) A point estimate of θ was denoted by θ̂ or θ̂(X) (with suitable subscripts to indicate the type of estimator or the method used to derive it). On the other hand, we used the notation C or C(X) = [L(X), U(X)] to denote an interval estimate of θ. But whether we consider point or interval estimation of θ, it is always a tricky issue to evaluate the performance of an estimator. Traditionally, if θ̂ = θ̂(X) is a point estimate of θ, then we study the bias of θ̂ given as B(θ̂) =
( E(θ̂)−θ),
and the mean squared error (MSE) of θ̂ given as MSE(θ̂) = E [ (θ̂ − θ)2].
Both B(θ̂) and MSE(θ̂) are functions of θ. Similarly, if C = [L(X), U(X)] is an interval estimate of θ, then we study Probability Coverage (PC) of C given as P (C) = P
