Question 3 [16 marks] Consider two independent random samples X1, . .., X, and Y1, ..., I'm; and assume they fol- low Binomial distributions: Xi ~ Bino(ni, m) and Y; ~ Bino(n2, #2), where m and 72 are unknown parameters of these Binomial distributions. Let /(#1, #2) denote the log-likelihood function from these two samples combined. (a) Demonstrate that the log-likelihood for #1, 72 from these two samples is given by I(1, 12) = log(m) >X,+log(1-m) _(n-X;)+log(#2) _Y, +log(1-#2) _(n2-Y;). 1= 1 j=1 (b) Using the above log-likelihood to show that the maximum likelihood estimators (MLEs) of m and 12 are 71 = nn1 mn2 (c) Derive the MLE for the ratio of m over 72, namely 0 = 21/72. (d) Next, you are required to perform a Wald test on the hypothesis Ho : 1 = #; and 12 = 72, where a; and ng are two known values. (i) Find the expected information matrix for * and #2- (ii) Write down the asymptotic normal distributions for the MLEs #, and #2. (iii) Explain how to perform the Wald test for this hypothesis testing problem. Your answer should include: the test statistic formula and how to determine a critical value