question 4 thank you

Answer all questions, and write your answers in a clear and concise manner. Calculators are not needed. Let (X1, Xa, . . .,Xin) denote a random sample from the density function f(I10) = -0" In(0)1(1 > 0), where e E (0, 1) is an unknown parameter. Denote N, to be the number of observations among the sample elements which belong to the interval (j-1, j],j = 1, 2and N3 = n-Ni-N2. For confidentiality, only summaries of the sample are released to two analysts, such that Analyst A knows only (N1, N2, Na) and Analyst B knows only (N1, N2 + N3)- Q1. Show that (N1, N2, Na) follows a multinomial distribution with probabilities 1 - 0, 0(1 - 0) and 02, respectively (s marks). Note that a random variable (X, Y, Z) follows a multinomial distribution with size n and proba- bilities p, q, and r = 1 - p - qif X + Y + Z = n and their joint p.m.f. is P(X = a, Y = b, Z = c) - - p q'r' n! for all non-negative integers a, b, c such that a + b + c = n. Q2. Based on the complete sample (X1, X2, . .. , Xn.), (a) Write down an expression for the likelihood function of 0 (2 marks). (b) Derive a scalar sufficient statistics for 0 (2 marks). (c) Find the Fisher Information about 0 (4 marks). Note that X follows some exponential distri- bution. (d) Find the MLE (Maximum Likelihood Estimator) of 0 and state its large-sample distribution (s marks). Q3. Based on the data released to Analyst A, (a) Show that the likelihood function of 0 is given by (1 - () NitN20 N2+2Ns (3 marks). (b) Find the Fisher information about 0 (4 marks). (c) Find the MLE of 0 and state its large-sample distribution (s marks). Q4. Based on the data released to Analyst B, (a) Write down an expression for the likelihood function of 0 (3 marks), (b) Find the Fisher information about @ (3 marks). (c) Show that the MLE of 0 is both unbiased and efficient (6 marks)