Page 360 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 28: Theory of Estimation: Point Estimation, Unbiasedness, Consistency, Efficiency and Sufficiency

μ p = P or E (P) = P Notes
(2) Consistent Estimator: An estimator is said to be consistent if the estimator approaches the
ˆ
population parameter as the sample size increases. In other words, an estimator θ is said to be
ˆ
consistent estimator of the population parameter θ , if the probability that θ approaches θ is
1 an n becomes large and larger. Symbolically,
( ˆ θ → ) θ P → 1 as →n ∞

Note: A consistent estimator need not to be unbiased
A sufficient condition for the consistency of an estimator is that
ˆ
θ
(i) E ( ) → θ
ˆ
θ
(ii) Var ( ) → 0 as n → ∞
Example 4: Sample mean X is a consistent estimator of the population mean μ because the
expected value of the sample mean approaches the population mean and the variance
of the sample mean approaches zero as the size of the sample is sufficiently increased.
Symbolically,

(i) () →EX μ

()
(ii) Var X = σ 2 → 0 as →n ∞
n
Example 5: Sample median is also consistent estimator of the population mean because:

(i) ( EM ) → μ
(ii) Var (M) → 0 as →n ∞
(3) Efficient Estimator: Efficiency is a relative term. Efficiency of an estimator is generally defined
ˆ
ˆ
by comparing it with another estimator. Let us to take two unbiased estimators θ and θ . The
2
1
ˆ
ˆ
estimator θ is called an efficient estimator of θ if the variance of θ is less than the variance
1
1
ˆ
of θ . Symbolically,
2
( Var ˆ 1 ) θ < ( Var ˆ 2 ) θ
ˆ
Then, θ is called an efficient estimator.
1
Example 6: Sample mean X is an unbiased and efficient estimator of the population mean (or
true mean) than the sample median M because the variance of the sampling distribution
of the means is less than the variance of the sampling distribution of the medians.
The relative efficiency of the two unbiased estimators is given below:
()
⋅
We know that, Var X = σ 2 , Var (M) = π σ 2
n 2 n

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