The most important target of Statistics is to draw inferences about a population from the analysis of a sample drawn from that population. The theory of estimation was set up by Prof. R. A. Fisher.
What is a parameter space?
Consider X = random variable with p.d.f f (x ,θ ), where θ∈Θ. The set Θ, which is the set of all possible values of θ, is called the parameter space.
Characteristics of Estimators:
A good estimator should satisfy the following criteria:
Let us now explain briefly the first criteria Consistency estimator:
An estimator is said to be sufficient for a parameter, if it contains all the information in the sample regarding the parameter.
If S= s(x1, x2.... xn) is an estimator of a parameter θ, based on a sample x1, x2.... xn of size n from the population with density function f (x, θ) such that the conditional distribution of x1, x2.... xn given S is independent of θ, then S is sufficient estimator for θ.
Theorems Based on Sufficiency:
The necessary and sufficient condition for a distribution to provide sufficient statistic is provided by the following theorem:
Factorization Theorem (Neymann): S = s(x) is sufficient for θ iff the joint density function M (say), of the sample values can be symbolically written as:
M = gθ [s(x)] h(x)
Where gθ [s(x)] h(x) depends on θ) and x only through the value of s(x) and h(x), which is independent of θ).
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