Page 371 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes
Again, since x , x , ... x is a random sample and μ is the population mean, we have ( E x − μ 2 = σ 2 .
1 2 n ) i
Therefore,

(
n E x − μ ) i 2 ∑ σ 2
2
σ
E ( ) = ∑ = = σ 2
0
i = 1 n n
Thus, σ 0 2 is an unbiased estimator of σ 2 .
Example 3: Find the m.l.e. of the parameters μ and σ 2 in random samples from a ( μ σN , 2 )
population, when both the parameters are unknown.
Solution: As in the preceding example,

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

∂ ⎡ ⎤ log L −1
∴ ⎢ ⎥ = ∑ ( −2 x 0 ) i ( μ 1 )− = 0
⎣ ∂μ ⎦ μ=μ 0 2 σ 2

∑
This gives ( i − x μ 0 ) = 0; i.e., μ = x , the sample mean. The m.l.e. of the parameter μ is the sample
0
mean x . (Note that this estimator is unbiased).

∑ ( − x μ ) i 2
2
Proceeding as in Example 2 we have σ = n . But since the parameter μ is not known, it is
0
replaced by the m.l.e. μ = x . The m.l.e. of σ 2 is now
2
∑ ( − x x ) i 2
σ 0 2 = n = S 2

which is the sample variance. (Note that this estimator is biased).
Example 4: A tossed a biased coin 50 times and got head 20 times, while B tossed it 90 times and got
40 heads. Find the maximum likelihood estimate of the probability of getting head when the coin is
tossed.
Solution: Let P be the unknown probability of obtaining a head. Using binomial distribution,
Probability of 20 heads in 50 tosses = 50 C P 20 ( 20 1 ) − P 30

Probability of 40 heads in 90 tosses = 90 C P 40 ( 40 1 ) −P 50
The likelihood function is given by the product of these probabilities:

L= 50 C 20 ⋅ 90 P 30 ( 40 1 P ) − C 30

∴ log L = ( log 50 20 90 C 40 ) + C + 60logP8 (0log 1P ) −

∂ logL 60 80
Hence, ∂ P = P − 1 − P
The maximum likelihood estimate P is therefore obtained by solving
0

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