ass3 - 2/09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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2/09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Assignment 3 1. For a distribution with mean μ 6 = 0 and standard deviation σ , its coefficient of variation c v is defined as c v = σ/μ . Let X 1 and X 2 be positive random variables independently drawn from a distribution with an unknown mean μ and known coefficient of variation c v . Consider two estimators of μ : T 1 = X 1 + X 2 2 and T 2 = X 1 + X 2 2 + c 2 v . (a) Find the bias and mean squared error (MSE) of T 1 . Is T 1 unbiased? (b) Find the bias and MSE of T 2 . Is T 2 unbiased? (c) Which of the two estimators would you recommend for estimating μ ? State clearly your criterion. 2. A source emits a signal Z which has a normal distribution with mean μ and variance σ 2 , so that the likelihood function based on Z = z is given by the function ψ ( μ, σ | z ) = 1 2 πσ 2 exp - ( z - μ ) 2 2 σ 2 . A detector observes the signal as Y = | Z | and fails to record its sign. (a) By considering the cdf of Y or otherwise, show that the pdf of Y is f Y ( y | μ, σ ) = ψ ( μ, σ | y ) + ψ ( - μ, σ | y ) , y > 0 . (b) Suppose μ = 0 and σ is unknown. (i) Denote by z ( σ ) and * y ( σ ) the likelihood functions based on Z = z and Y = y respectively. Show that z and * y are essentially the same. [ Hint: show that both likelihood functions are proportional to ψ (0 , σ | y ) . ] (ii) Show that the Fisher information contained in Z about σ is 2 σ - 2 . (iii) What is the Fisher information contained in Y about σ ? Is there any information loss due to the failure to record the sign of Z ? 1
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(iv) Do you think Y is sufficient for σ ? Why? (c) Suppose σ = 1 and μ is unknown. (i) Show that the likelihood function based on Y = y is ** y ( μ ) = ψ ( μ, 1 | y ) + ψ ( - μ, 1 | y ) . (ii) Show that the Fisher information contained in Z about μ is 1. (iii) Show that the Fisher information contained in Y about μ is 1 - 4 E h Y 2 ( e μY + e - μY ) - 2 i . Is there any information loss due to the failure to record the sign of Z ? 3. Let X 1 , . . . , X n be independent Poisson random variables with X j having parameter , where λ > 0 is an unknown parameter. (a) Find the Fisher information contained in ( X 1 , . . . , X n ) about λ .
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