Question

Problem 6. Gamma distribution is defined by its density function T(a) when t 0 and 0 otherwise. Here, θ is unknown parameter, a is a known param- eter Г(a) is a normalization constant (like in chi-square distribution discussed in class, see also p. 99 of the textbook) a) Check that Gamma distribution can be written in the exponential form. b) Use this fact to give a sufficient statistic for θ (you can do it with no additional computations after part a c) Give examples of two more different sufficient statistics for θ.

Answer 1

Given gamma distribution is, f(t, theta, a) = theta^a/gamma (a) t^a - 1 e^- theta t, t greaterthanorequalto 0 0, otherwise i) The gamma distribution can be written in exponential for if we can write PDF F(yi, theta, phi) as f(y_i, theta, phi) = exp [y_i theta_i - b (theta_i)/a (phi) + c (y_i, phi)] where v, are independent random variables & phi is a scale parameter & phi is a scale parameter it is clear obviously that this can be done for the gamma distribution f (t, theta, a) = e^log [theta^a t^a - 1 e^b/gamma (a)] = exp [log theta^a + log t^a - 1 - (log gamma (a) + theta t)] = exp [a log theta + (a - 1) log t - log Gamma (a) - theta t] which is in the exponential form ii) pi^n_i = 1 f(t_n theta, a) = exp sigma^n_i = 1 [a log theta + (a - 1) log t_1 - log Gamma (a) - theta (t_2) clearly, iw we take