Likelihood Questions - Example 10:15 Given x...

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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.

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