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03_Estimation_part5

03_Estimation_part5 - Ch8 p.61 Statistics like R are called...

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NTHU MATH 2820, 2008, Lecture Notes Ch8, p.61 Question 6.6 Note that minimal su cient statistics may still contain ancillary informa- tion. What other property can guarantee su cient statistics containing no ancillary information? Statistics like R are called ancillary statistics , which have distri- butions free of the parameters and seemingly contain no information about the parameters. (other example of ancillary statistics?) Definition 6.17 (completeness, TBp.310) Let f ( s | θ ), θ , be a family of pdfs or pmfs for a statistic S = S ( X 1 , . . . , X n ). The family of probability distributions is called complete if E[ u ( S )] = 0 for all θ implies u ( S ) = 0 with probability 1 for all θ . Equivalently, S is called a complete statistics. T = g ( X 1 , . . . , X n ) S = h ( T ) u 1 ( S ): a non-constant function u 2 ( S ): a constant function made by Shao-Wei Cheng (NTHU, Taiwan) Ch8, p.62 Example 6.28 (sufficient and complete statistics of i.i.d. Uniform distribution U( 0 , θ ) ) Let X 1 , . . . , X n be i.i.d. from Uniform distribution U (0 , θ ), θ > 0. By factorization theorem, X ( n ) , the largest order statistics, is su cient. The pdf of X ( n ) is nx n 1 θ n I (0 , θ ) ( x ) . Let u be a function such that E [ u ( X ( n ) )] = 0 for all θ . Then θ 0 u ( x ) x n 1 dx = 0 , for all θ > 0, which implies u ( x ) x n 1 = 0 , a.s. for x (0 , ), therefore, X ( n ) is complete. Note. If S is a complete statistic, E [ u ( S )] is a constant for all θ implies that the trnasformation u is a constant transformation. If S is a complete statistic, any transformations of S except the constant functions contains some information about θ . In Example 6.27, E ( R ) = n 1 n +1 . That is, X ( n ) X (1) n 1 n + 1 = R E ( R ) has mean zero for all θ . there is a nonzero function of X (1) and X ( n ) whose expectation is zero for all θ
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NTHU MATH 2820, 2008, Lecture Notes Ch8, p.63 Example 6.29 (sufficient and complete statistic of i.i.d. Poisson distribution) Theorem 6.14 Definition 6.18 (one-parameter exponential family of probability distributions, TBp.308) Suppose X 1 , . . . , X n is an i.i.d. sample from Poisson distribution P ( λ ). Then f ( x 1 , . . . , x n ; λ ) = ( e n λ λ n i =1 x i ) / n i =1 x i ! . So S = n i =1 X i is su cient for λ and S P ( n λ ). If u ( S ) is any function of S s.t., 0 = E[ u ( S )] = e n λ s =0 u ( s )( n ) s s ! λ s , for all λ , then, all coe cients of λ are zero and u ( s ) = 0. Hence S is also complete. A complete and su cient statistic is minimal su cient. However, a minimal su cient statistic is not necessarily complete. (For example, the minimal su cient statistic ( X (1) , X ( n ) ) in Ex. 6.26 is not complete because E ( X (1) ) = 1 n +1 + θ , E ( X ( n ) ) = n n +1 + θ , and E ( X ( n ) X (1) n 1 n +1 ) = 0, for any θ > 0.) A family of distributions { f ( x | θ ) : θ } is a one-parameter exponential family if the pdf or pmf is of the form: f ( x | θ ) = exp [ c ( θ ) T ( x ) + d ( θ ) + S ( x )] = e c ( θ ) T ( x ) e d ( θ ) e S ( x ) , x A 0 , x / A where the set A does not depend on θ .
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