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Let Y₁... Yn denote a random sample from the binomial distribution Bin(m, n), where is

an unknown parameter.

(a) Find the maximum likelihood estimator (MLE) of . (You must provide the likelihood

function and show the detailed process of deriving the MLE.)

(b) Find the expected information and then calculate the asymptotic standard error for

the MLE in (a).

(c) Consider testing the hypothesis that the parameter is equal to a particular value:

Ho π = 0.6 versus H₁ : 0.6.

Write down the test statistics corresponding to each of the following hypothesis testing

methods and also evaluate them under the assumption that n = 30 and the observed

value of the MLE is = 0.5:

(i) likelihood ratio test

(ii) score test

(iii) Wald test

(iv) approximate Wald test.