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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II Assignment 5 (Sketch Solution) 1. Let ˆ θ be the mle and s . e . ( ˆ θ ) be the s.e. of ˆ θ . Largesample theory gives that ( ˆ θ θ ) / s . e . ( ˆ θ ) ∼ N (0 , 1) approximately. To test H : θ = θ vs H 1 : θ 6 = θ at 5%, reject H if fl fl fl ( ˆ θ θ ) / s . e . ( ˆ θ ) fl fl fl > 1 . 96 . A 95% confidence interval is n θ : fl fl fl ( ˆ θ θ ) / s . e . ( ˆ θ ) fl fl fl ≤ 1 . 96 o = ˆ θ ± 1 . 96s . e . ( ˆ θ ) . (a) The likelihood is ‘ ( θ ) = θ n e ∑ i X i /θ , which is max’d at the mle ˆ θ = ¯ X . Note that i ( θ ) ={ E [ ∂ 2 ‘ ( θ ) /∂θ 2 ] } = n/θ 2 and we can take s . e . ( ˆ θ ) = ¯ X/ √ n . (b) The likelihood is ‘ ( θ ) = θ 2 n ( Q i X i ) e θ ∑ i X i , which is max’d at θ = ˆ θ = 2 / ¯ X . As in (a), we find i ( θ ) = 2 n/θ 2 and so s . e . ( ˆ θ ) = ˆ θ/ √ 2 n . (c) The likelihood is ‘ ( θ ) = e nθ θ ∑ i X i / Q i X i !, which is max’d at θ = ˆ θ = ¯ X . As in (a), we find i ( θ ) = n/θ and so s . e . ( ˆ θ ) = ( ¯ X/n ) 1 / 2 . 2. (a) P ( X 1 /λ ≤ u ) = R λu f ( x  λ ) dx = 1 e u . (b) (a) implies P ( X 1 / ln α ≤ λ ) = 1 α . Thus we may take λ α ( x 1 ) = x 1 / ln α , which is a (1 α ) lower confidence bound for λ . 3. Note that the mle of θ is ˆ θ = X/n . If n is large, ( X nθ ) / p X (1 X/n ) ∼ N (0 , 1) ap proximately. Find k α such that P ( N (0 , 1) ≥ k α ) = 1 α . Then we may take u α ( X ) = X/n ( k α /n ) p X (1 X/n ). Illustration: we have k . 1 = 1 . 28. Substitute this into above to get the answer. 4. Note that P (  X θ  ≤ c ) = R c c e y  / 2 dy = 1 e c . Equating this to 1 α gives c = ln α . Thus the required C.I. is [ X + ln α,X ln α ]. 5. (a) First we see that the model is mlr in ∑ i T i and a suitable test is to use a critical region { ∑ i T i > k } = { ∑ i T i < k } . (In fact, this test is UMP.) Determine k according to P ( X i T i < k  λ ) =...
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This note was uploaded on 04/15/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.
 Spring '10
 SMSLee
 Statistics, Probability

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