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ST 702

David Elsheimer Inglés 9 fichas

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1What is a sufficient statistic, and what is the factorization theorem?A statistic T(X) is called a sufficient statistic for θ if the conditional distribution of X given T(X) does not depend on θ.
Factorization theorem is that f(X|θ) = g(T(X)|θ)h(X)
2Define a min. suff statisticA suff stat is min suff if for any suff stat a min suff stat is a function of it. Can be determined if a statistic is min suff using Lehmann-scheffe (f(X|θ)/f(Y|θ) is constant as a function of theta)
3Define an ancillary statisticAn ancillary statistic doesn't depend on θ
4Define a complete statisticA statistic is complete if E(g(T)) = 0 for all θ implies P(g(T) = 0) = 1 for all θ
Any function of a complete stat is complete
5Explain how an exponential family can be shown easily whether or not it's complete.If the k w_j(θ)'s can be expressed as an open set in R^k, then joint statistic of the sums of the t_j(X)'s is complete
6What is a cramer-rao lower bound, and how is attainment verified?For an estimator W(X), the lower bound is (d/dθW(X))^2/(E(d/dθ(log f(X|θ))^2)
Assuming that observations are iid, the numerator (I_N(θ)) is nI(θ), and if it's from a one-parameter exp. fam, leibniz rule can be applied and (E(d/dθ(log f(X|θ))^2) = -(E(d^2/dθ^2(log f(X|θ))^2)
7Define a bayes estimatorπ(p|X) = π(p)f(X|p)/m(x)
8Basu's theoremA comp. stat and an ancillary stat must be independent
9What is MSE(W)var(W) + Bias(W)^2