Assignment 5

Assignment 5 - 4/2010 THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: 4/2010 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Assignment 5 1. Take a random sample X 1 ,...,X n from the distribution f ( · | θ ) given below. Quoting the large- sample theory of the mle, construct a size 5% (Wald’s) test of H : θ = θ against H 1 : θ 6 = θ , and hence derive an approximate 95% confidence interval for θ . (a) f ( x | θ ) = θ- 1 e- x/θ , x > 0; (b) f ( x | θ ) = θ 2 xe- θx , x > 0; (c) f ( x | θ ) = e- θ θ x /x !, x = 0 , 1 , 2 ,... . 2. Let X 1 be a random variable having the pdf f ( x | λ ) = λ- 1 exp(- x/λ ), x > 0. Let x 1 be a realisation of X 1 . (a) Show that P ( X 1 /λ ≤ u ) = 1- e- u , u > . (b) Using (a), show how to find λ α ( x 1 ) such that P { λ α ( X 1 ) ≤ λ } = 1- α, for all λ > 0. How would you call the statistic λ α ( x 1 )? 3. Suppose X ∼ Binomial( n,θ ). Show how to construct u α ( X ) such that P { ≤ θ ≤ u α ( X ) } ≈ 1- α, for all θ , if n is large. Illustrate your answer for n = 100, x = 10, α = 0 . 1. [Hint: For large n , the statistic X- nθ p X (1- X/n ) is approximately normal with mean 0 and variance 1.] 4. Suppose f ( x | θ ) = 1 2 exp {-| x- θ | ) } ,-∞ < x < ∞ . For a single observation from this pdf, construct a 100(1- α )% equal-tailed confidence interval for θ . [Hint: Consider the probability P ( | X- θ | ≤ c ) for an observation X drawn from f .] 1 5. Consider a random sample ( T 1 ,...,T n ) drawn from an exponential distribution with rate λ > 0. It can be shown that 2 λ ∑ n i =1 T i has a chi-squared distribution with 2 n degrees of freedom. (a) Derive a size 0.05 critical region for the likelihood ratio test of H : λ = λ against H 1 : λ > λ ....
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Assignment 5 - 4/2010 THE UNIVERSITY OF HONG KONG...

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