Page 359 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes 28.2 Unbiasedness, Consistency, Efficiency and Sufficiency

There can be more than one estimators of a population parameter. For example, the population mean

X
() μ may be estimated either by sample mean () or by sample median (M) or by sample mode (Z),
2
σ
2
etc. Similarly, the population variance ( ) may be estimated either by the sample variance (s ),
sample S.D. (s), sample mean deviation, etc. Therefore, it becomes necessary to determine a good
estimator out of a number of available estimators. A good estimator is one which is as close to the
true value of the parameter as possible. A good estimator possess the following characteristics or
properties:
(1) Unbiasedness
(2) Consistency
(3) Efficiency
(4) Sufficiency
Let us consider them in detail:
ˆ
(1) Unbiased Estimator: An estimator θ is said be unbiased estimator of the population parameter
ˆ
θ if the mean of the sampling distribution of the estimator θ is equal to the corresponding
population parameter θ . Symbolically,

μ ˆ = θ
θ
ˆ
In terms of mathematical expectation, θ is an unbiased estimator of θ if the expected value of
the estimator is equal to the parameter being estimated. Symbolically,
ˆ
θ
( ) = θ
E
Example 1: Sample mean X is an unbiased estimate of the population mean μ because the mean
EX
of the sampling distribution of the means μ or () is equal to the population
X
mean μ . Symbolically,
μ = μ or () = μ
EX
X
Example 2: Sample variance s is a biased estimate of the population variance σ 2 because the
2
mean of the sampling distribution of variance is not equal to the population variance.
Symbolically,
2
μ ≠ σ 2 or ( E s 2 ) ≠ σ 2
s
2
ˆ s
However, the modified sample variance ( ) is unbiased estimate of the population
variance σ because
2

( E s ˆ 2 ) = σ 2 where, ˆ s 2 = n n − 1 × s 2

Example 3: Sample proportion p is an unbiased estimate of the population proportion P because
the mean of the sampling distribution of proportion is equal to the population
proportion. Symbolically,

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