Unbiased 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:

Unbiased Estimator:

A statistic used to evaluate a population parameter is unbiased if the mean of the sampling distribution of the statistic is equal to the true value of the parameter being evaluated. This is a property related with finite n. In other words, a statistic = (x₁, x₂.... x_n) is said to be an unbiased estimate of parameter y if

E ( ) = y

If E () > y then is said to be positively biased whereas if E () < y, then is said to be negatively biased. The amount of bias b(y) is given by

b(y) = E ( ) -  y

Let us consider a sampling from a population with mean μ and variance σ². Then, we know that,

E()=μ and E( s² ) ≠ σ² but E( s² ) = σ²

Therefore, there is a reason to prefer s² = 1/ (n - 1) [ ∑ (x_i - )²] where I = 1 to n to the sample variance

s2 = 1/ n ∑ (x_i - )2 where I = 1 to n.