Efficiency 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:

Efficiency Estimator:

Even if we restrict ourselves to unbiased estimates, there will, in common, exist more than one consistent estimator of a parameter. Thus, there is a requirement of some further criterion which will make us to select between the estimators with the common property of consistency. Such a criterion, based on the variances of the sampling distribution of estimators is generally called as efficiency.

If Θ₁ is the most efficient estimator with variance v1 and Θ₂ is any other estimator with variance v2, then the efficiency E of Θ₂ is defined as

E = v₁ / v₂

In other words, an estimator is said to be efficient if it estimates or calculates the parameter of interest in some best possible fashion.

Consider the following example:

In the normal samples since sample mean x bar is the most efficient estimator for

Efficiency Estimator:

Even if we restrict ourselves to unbiased estimates, there will, in common, exist more than one consistent estimator of a parameter. Thus, there is a requirement of some further criterion which will make us to select between the estimators with the common property of consistency. Such a criterion, based on the variances of the sampling distribution of estimators is generally called as efficiency.

If Θ₁ is the most efficient estimator with variance v1 and Θ₂ is any other estimator with variance v2, then the efficiency E of Θ₂ is defined as

E = v₁ / v₂

In other words, an estimator is said to be efficient if it estimates or calculates the parameter of interest in some best possible fashion.

Consider the following example:

In the normal samples since sample mean x bar is the most efficient estimator for μ, the efficiency of Median for such samples (for large n) is μ , the efficiency of Median for such samples (for large n) is

E = V () / V (Median) - (σ² / n) / (πσ²/ 2n) - 2 /π - 0.637