(b) Recall that X follows a Poisson distribution with mean > > 0 if P(X =...

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(b) Recall that X follows a Poisson distribution with mean > > 0 if P(X = k) = e for k 0, 1, 2.... Now let X1,..., Xm be a sample from the Poisson distribution with (unknown) mean , and Y,..., Y be a sample from the Poisson distribution with (unknown) mean +A. Furthermore, {X1,..., Xm, Y,..., Yn} are independent of each other. i. Find the estimators for and A using method of moments. Are they consistent (as n, m)? Now find the standard errors for the estimators. ii. Find the maximum likelihood estimators for and A. iii. Suppose that m = 40, Xm = 1 X = 10, n = 100 and Yn = n = 11. m Would you reject Ho: A = 0 against H: A0 at the 5% significance level? Would you reject Ho A = 0 against H A > 0 at the 5% significance level? : Briefly comment on your findings. [10 marks for both parts; 20 marks in total] (b) Recall that X follows a Poisson distribution with mean > > 0 if P(X = k) = e for k 0, 1, 2.... Now let X1,..., Xm be a sample from the Poisson distribution with (unknown) mean , and Y,..., Y be a sample from the Poisson distribution with (unknown) mean +A. Furthermore, {X1,..., Xm, Y,..., Yn} are independent of each other. i. Find the estimators for and A using method of moments. Are they consistent (as n, m)? Now find the standard errors for the estimators. ii. Find the maximum likelihood estimators for and A. iii. Suppose that m = 40, Xm = 1 X = 10, n = 100 and Yn = n = 11. m Would you reject Ho: A = 0 against H: A0 at the 5% significance level? Would you reject Ho A = 0 against H A > 0 at the 5% significance level? : Briefly comment on your findings. [10 marks for both parts; 20 marks in total]