Page 370 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 29: Methods of Point Estimation and Interval Estimation

Solving, we get m = x . Again, Notes
0
∂ ⎡ 2 logL ⎤ nx nx
⎢ ⎥ = − 2 = − 2 = − n which is negative.
⎢ ∂ m 2 ⎦ ⎥ ⎣ m = m 0 m 0 x x

This shows that log L is maximum at m = m = x . That is the m.l.e. of m is m = x , the sample mean.
0 0
Example 2: Find the maximum likelihood estimator of the variance σ 2 of a Normal population
N ( μ σ, 2 ) , when the parameter μ is known. Show that this estimator is unbiased.

Solution: The p.d.f. of Normal distribution is

1 ( − x μ ) − 2 σ /2 2
fx 2 ) = e ; ( ∞ − < x + ∞ < )
( ,μ σ,
σ π 2
The logarithm (to the base e) of the likelihood function L is
n
(∑
,
log L = log fx , μ σ 2 ) i
i = 1
⎡ 1 ( x − μ 2 ⎤
= ∑ ⎢ ⎢ ⎣ σ −log 2 ( log )2x − 2 σ 2 ) i ⎥− ⎥ ⎦

n n ∑ ( − x μ ) i 2
= − σ − log 2 ( log )2x −
2 2 2 σ 2
Differentiating partially with respect to σ ,
2

∂ log L − n + ( ∑ x − μ ) i 2
( ∂σ 2 ) = 2 σ 2 ( 2 2 ) σ 2

The m.l.e. of σ 2 is obtained by solving

∑ ( n − x μ ) i 2
− + = 0
2 σ 2 2 σ 0 4

n ( x − μ ) i 2
∴ σ 0 2 = ∑
i = 1 n

⎡ ⎤
∂ ⎢ 2 log L ⎥ −n
It can be shown that ⎢ 2 ⎥ = 2 σ 4
⎢ ⎣ ( ∂σ 2 ) ⎥ σ=σ 2 0 0
2 ⎦

which is negative. Thus the maximum likelihood estimator of σ is
2

σ 0 2 = ∑ ( x − n μ ) i 2 , ( μ known)

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