Consistent 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:

Consistent Estimator:

Let be an estimator of y based on as sample of size n. Then {} is a consistent sequence of estimators of y (or is consistent for y) if

tends to y at p as n → ∞

That is,for every ε > 0,

or in other words,for every ε > 0, α > 0

P ( | – y | < ε) > 1 – α, n ≥ N

N=some very large value of n.

Consider the following example to learn about consistent estimator:

Let sample x₁, x₂.... x_n be a random sample from a Poisson population with parameter c. Consider

=1/n(x₁+x₂+x₃+….+x_n) as an estimator of c. As xi are i.i.d, P(c), E (x_i) = c. Hence by the Weak Law of Large Numbers),

This implies that,is a consistent estimator of c.