Unformatted text preview: Example 10:15. Given x “successes”
’in n trials, …nd the maximum likelihood estimate of the parameter of the corresponding binomial distribution.
Solution 10:15. To …nd the value of that maximises
L( ) = n
x x )n (1 x it will be convenient to make use of the fact that the value of
L( ) will also maximise
ln L( ) = ln n
+ x ln + (n
x x) ln(1 that maximises ): Thus, we get
xnx
d (ln L ( ))
=
;
d
1
and, equating this derivative to 0 and solving for ; we …nd that the likelihood
x
function has a maximum at = n : This is the maximum likelihood estimate
of the binomial parameter , and we refer to ^ = X as the corresponding
n
maximum likelihood estimator.
Example 10:16. If x1 ; x2 ; :::; xn are the values of a random sample from an
exponential population, …nd the maximum likelihood estimator of its parameter
:
Solution 10:16. Since the likelihood function is given by
(
)
n
n
n
Y
1
1X
L( ) = f (x1 ; x2 ; :::; xn ; ) =
f (xi ; ) =
exp
xi ;
i=1 i=1 di¤erentiation of ln L( ) with respect to yields
n d (ln L ( ))
=
d + n
1X
2 xi : i=1 Equating this derivative to zero and solving for ; we get the maximum likelihood
estimate
n
X
^= 1
xi = x:
n i=1
Hence, the maximum likelihood estimator is ^ = X : 1 ...
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This note was uploaded on 10/25/2011 for the course ECON 501 taught by Professor Zobuz during the Spring '11 term at Istanbul Technical University.
 Spring '11
 zobuz

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