Page 397 - DMTH404_STATISTICS

Unit 29: Estimation of Parameters: Criteria for Estimates

or X(1- p) - (n - X)p = 0 Notes

X
This gives ˆ  = , where   denotes an estimator of p.
n

d 2 log L X
It can also be shown that < 0 when ˆ  = .
d 2 n

Example 3: Obtain the maximum likelihood estimator of the parameter m of the Poisson
distribution.
Solution.

Let X , X , ...... X be a random sample of n independent observations from the given population.
1 2 n
Therefore, we can write
e - m .m X 1 e - m .m X 2 e - m .m X n e - nm .m å i X
L = ´ ´ ...... ´ =
X 1 ! X 2 ! X n ! Õ ( ) !X i

Taking logarithm of both sides, we get

logL = - nm + å X logm - å log ( ) !X
i i
Differentiating w.r.t. m, we get

d log L å X i å X i
= - n + = 0 Þ ˆ m = = X
dm m n
Thus, sample mean is MLE of parameter m.

2
Example 4: For a normal population with parameter  and  , obtain the maximum
likelihood estimators of the parameters.
Solution.
The probability density function of normal distribution is

1 - 1 (X - ) 2
)
f ( ; ,X   = e 2  2
 2
Given a random sample of n independent observations, the likelihood function L is given by
n
L = Õ ( f X i ; , ) .

i= 1
Taking logarithm of both sides, we get

2
æ 1 - 1 (X - ) ö 1 1
log L = å log ç e 2  2 ÷ = å log - 2 å (X - )  2
ç  2 ÷  2 2 i
è ø

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