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Unformatted text preview: 1 n i i W X = = ∑ and ˆ W p n = . (a). Please derive the method of moment estimator of p. (b). Please derive the maximum likelihood estimator of p. Solutions 2. (a) ( 29 ( 29 [ ] . . . 2 2 2 2 2 2 ~ ( ) , ( ) 1* 0*(1 ) ; ( ) 1 * 0 *(1 ) (1 ); i i d i i i i i X Bernoulli p E X p p p Var X E X E X p p p p p p p = +  = =  = +   =  =  Therefore the first population mean is ( ) i E X p = And the first sample mean is: 1 ˆ n i i X W p n n = = = ∑ Set them to be equal and we found the moment estimator of p to be: ˆ W p n = . 2. (b) [i] 1 ( ) (1 ) , 1, , i i x x i f x p p i n== L [ii] ( ) (1 ) i i x n x i L f x p p∑ ∑ = =∏ [iii] ln ( )ln ( )ln(1 ) i i l L x p n x p = = +∑ ∑ [iv] 1 i i x n x dl dp p p==∑ ∑ ˆ i X W p n n ⇒ = = ∑ is the MLE of p...
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This note was uploaded on 11/14/2010 for the course AMS 312 taught by Professor Zhu,w during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Zhu,W

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