Examples of a sufficient statistic I. Show that q(X) = El
Chapter , Problem 12(choose chapter or problem)
Examples of a sufficient statistic I. Show that q(X) = El Xi is a sufficient statistic in the following cases: (a) Xl , . . . , Xn are LLd. Bernoulli random variables with parameter 9. (b) Xl , . .. , Xn are LLd. Poisson random variables with parameter 9. Solution. (a) The likelihood function is so it can be factored as the product of the function 9Q(x) (1 - 9)UQ( x) , which depends on x only through q(x), and the constant function s(x) == 1. The result follows from the factorization criterion for a sufficient statistic. (b) The likelihood function is so it can be factored as the product of the function e-99Q(x) , which depends on x only through q(x), and the function s(x) = 1/ TI1 Xi !, which depends only on x. The result follows from the factorization criterion for a sufficient statistic.
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