tailieunhanh - A Course in Mathematical Statistics phần 6

Tham khảo tài liệu 'a course in mathematical statistics phần 6', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 262 11 Sufficiency and Related Theorems each one of the n different ways in which the t successes can occur. Then if there are values of 9 for which particular occurrences of the t successes can happen with higher probability than others we will say that knowledge of the positions where the t successes occurred is more informative about 9 than simply knowledge of the total number of successes t. If on the other hand all possible outcomes given the total number of successes t have the same probability of occurrence then clearly the positions where the t successes occurred are entirely irrelevant and the total number of successes t provides all possible information about 9. In the present case we have P9X Xi . Xn xn T t P9IXi Xi . Xn Xn I T t -i 9 11 n n Ì p9 t t Pẹ X1 x1 . Xn Xn Pg T t if X1 . . . Xn t and zero otherwise and this is equal to 9x1 1 -9 1-X1 . . . 9x 1 -9 1-x n Í . _ n-i n 9 1 - 9 V17 n-t 9 1 - 9 n . n-t n 9 1 - 9 V17 1 c n V17 if X1 xn t and zero otherwise. Thus we found that for all x1 . xnsuch that xj 0 or 1 j 1 . n and n y xj t p0 X1 x1 . j 1 Xn Xn I T t V 1 7 independent of 9 and therefore the total number of successes t alone provides all possible information about 9. This example motivates the following definition of a sufficient statistic. DEFINITION 1 LetXj j 1 . nbe . . s with . 0 0 91 . 9 e Qc Rr and let T T1 . Tm where Tj Tj X1 . Xn j 1 . m are statistics. We say that T is an m-dimensional sufficient statistic for the family F 0 0 e Q or for the parameter 0 if the conditional distribution of X1 . Xn given T t is independent of 0 for all values of t actually for almost all . t that is except perhaps for a set N in Rm of values of t such that P0 T e N 0 for all 0 e Q where Pg denotes the probability function associated with the . 0 . REMARK 1 Thus T being a sufficient statistic for 9 implies that every measurable set A in nr P0 X1 . Xn e AIT t is independent of 0 for . Sufficiency Definition and Some Basic Results