STAB57H3 Lecture Notes - Lecture 28: Independent And Identically Distributed Random Variables, Joint Probability Distribution, Likelihood Function
Document Summary
Verify that t = p3 i=1 xi is a sufficient statistic for. T(x1, x2, xn) is said to be sufficient for if the joint probability function factors in the form f (x1, x2, xn|) = g[t(x1, x2, xn),] h(x1, x2, xn) where, H(x1, x2, xn) is a function of sample observations only. G[t(x1, x2, xn), involves and the sufficient statistic t. Note: by now we know f (x1, x2, xn|) is just the likelihood function. Proof of this theorem is not needed for the course. Therefore, according to the factorization theorem, t = pni=1 xi is a sufficient statistic for. Note: when we maximize likelihood, we maximize g[t(x1, x2, xn),. Hence, mle is a function of sufficient statistic t(x1, x2, xn). A sufficient statistic makes a reduction to the data. A sufficient statistic t for a model is a minimal sufficient statistic whenever the value of t(s) can be calculated once we know the likelihood function l(. |s).
