Question

4. Given a sample of size of moments to estimate the probabilities vector p from a multinomial (m; pi,.. . ,p,) distribution use the ,Pr) assuming m is method o known - (pi,

Answer 1

Multinomial (m: p_1,...pr) distribution probability function of the multinomial is p(x_1,x_2,..x_k) = n!/x_1 ! x_2 ! - x_k ! p^x_1_1 p^x_k_k , 0 lessthanorequalto x_i lessthanorequalto n = n!/Intersection^k_l = 1 x_i ! . Intersection^k_i = 1 pi^xi = n (p_1 + p_2 + ...p_k)^n , sigma^k_i = 1 p_i = 1. Moments of multinomial Distribution: moment generating function given by M_x (t) = M_x_1,x_2,...x_k (t_1,t_2,...t_k) = E {exp {sigma^k_i = 1 t_i x_i}} = sigma_x [n!/x_1 ! x_2 !...z_k ! (p_1 e^t_1)^x_1 (p_2 e^t_2)^x_2....(p_k e^tk)^x_k] = (p_1 e^t_1 + p_2 e^t_2 + ... + p_k e^t_k)^n where x = (x_1,x_2....x_k) M_x_1 (t_1) = M_x (t_1,0,0,..0) - (p_1 e^t_1 + p_2 + p_3 + ... + p_k)^n = [(1 - p_1) + p_1 e^t_1]^n x_1 backsim B (n_1 p_1) similarly x_i backsim B(n,p_i): I = 1,2,..k \r\n E(x_i) = npi & var (x_i) = npi (1 - pi) I = 1,2.. Here n