Sufficient Estimator

Introduction:

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.

Definitions:

• Statistic - Any function of the random sample x₁, x₂.... x_n that are being observed, say P_n (x₁, x₂x_n) is termed as a statistic. Precisely, a statistic is a random variable.
• An Estimator - If a statistic is used to estimate an unknown parameter a of the distribution, it is termed as an estimator.
• An Estimate - A particular value of the estimator, say, P_n (x₁, x₂....x_n) is called an estimate of α.

Characteristics of Estimators:

A good estimator should satisfy the following criteria:

• Unbiased ness
• Consistency
• Efficiency
• Sufficiency

Let us now explain briefly the first criteria Consistency estimator:

Sufficient Estimator:

An estimator is said to be sufficient for a parameter, if it contains all the information in the sample regarding the parameter.

Definition:

If S= s(x₁, x₂.... x_n) is an estimator of a parameter θ, based on a sample x₁, x₂.... x_n of size n from the population with density function f (x, θ) such that the conditional distribution of x₁, x₂.... x_n 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).