Estimator

In Statistics , an estimator is a Function of the known sample data that is used to estimate an unknown population Parameter ; an ''estimate'' is the result from the actual application of the function to a particular Set of data. Many different estimators are possible for any given parameter. Some criterion is used to choose between the estimators, although it is often the case that a criterion cannot be used to clearly pick one estimator over another.
To estimate a parameter of interest (e.g., a population mean, a binomial proportion, a difference between two population means, or a ratio of two population standard deviation), the usual procedure is as follows:

1- Select a random sample from the population of interest.

2- Calculate the point estimate of the parameter.

3- Calculate a measure of its variability, often a Confidence Interval .

4- Associate with this estimate a measure of variability.

There are two types of estimators: Point Estimator s and Interval Estimator s.

POINT ESTIMATORS

For a point estimator

\widehat{	heta}

of parameter

heta

,

# The '' Error '' of

\widehat{	heta}

\widehat{	heta} - 	heta

# The '' Bias '' of

\widehat{	heta}

is defined as

B(\widehat{	heta}) = \operatorname{E}(\widehat{	heta}) - 	heta.

\widehat{	heta}

is an '' Unbiased Estimator '' of θ Iff

B(\widehat{	heta}) = 0

for all θ, or, equivalently, iff

\operatorname{E}(\widehat{	heta}) = 	heta

for all θ.
# The ''mean squared error'' of

\widehat{	heta}

is defined as

\operatorname{MSE}(\widehat{	heta}) = \operatorname{E}[(\widehat{	heta} - 	heta)^2].

\operatorname{MSE}(\widehat{	heta}) = \operatorname{var}(\widehat	heta) + (B(\widehat{	heta}))^2,

:i.e. mean squared error = variance + square of bias.

where var(''X'') is the Variance of ''X'' and E(''X'') is the Expected Value of ''X''.

The Standard Deviation of an estimator of θ (the Square Root of the variance), or an estimate of the standard deviation of an estimator of θ, is called the '' Standard Error '' of θ.

CONSISTENCY

A consistent estimator is an estimator that Converges In Probability to the quantity being estimated as the sample size grows.

An estimator

t_n

(where ''n'' is the sample size) is a consistent estimator for Parameter

heta

if and only if, for all

\epsilon > 0

, no matter how small, we have

:

\lim_{n o\infty}{ m Prob}\left\{

Information About

Estimator