Assignment3 - 2/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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2/2011 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 | ) = ψ ( | y ) + ψ ( - | 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 | 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 loss in Fisher information 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 loss in Fisher information 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 n ) about λ . (b) Find the mle of λ . What are its bias, variance and mean squared error? 4. For each of the following parametric models, calculate the Fisher information about θ contained in a random variable X distributed under the model, as well as the Fisher information contained in an i.i.d. random sample of n observations drawn from that model: (a) about θ = ln λ under a Poisson distribution with mean λ ; [Hint: consider the loglikelihood function of θ based on a Poisson ( e θ ) random variable.] (b) about θ = σ 2 under N (0 2 ); (c) about θ = p under a negative binomial distribution with mass function f ( x ) = ˆ m + x - 1 m - 1 ! p m (1 - p ) x , x = 0 , 1 , 2 ,... . 5. (a) For the scale family with density (1 ) f ( x/θ ), θ > 0, show that the Fisher information a single observation X has about θ is 1 θ 2 Z ‰ y f 0 ( y ) f ( y ) + 1 ± 2 f ( y ) dy.
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.

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Assignment3 - 2/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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