辅导案例-STAT0008
STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 6 Q2. The random variables X1, . . . , Xn are independent and sampled from a distribution with probability density function f(x;ψ) = 1 ψ exp ( 1 ψ − 1 ψ ex + x ) x ≥ 0; 0 otherwise; where ψ > 0 is an unknown parameter. (a) Determine the maximum likelihood estimator of ψ and call this estimator ψˆ. [3] (b) Is ψˆ is the minimum variance bound unbiased estimator (MVBUE) of ψ? Justify your answer. [3] (c) Determine the Crame´r-Rao lower bound for the variance of unbiased estimators of 1/ψ. [3] (d) Determine the median of the distribution of Xi, for any i ∈ {1, . . . , n}. [6] A sample of X1, . . . , Xn, denoted x1, . . . , xn, is taken where n = 60, ∑60 i=1 xi = 45.95 and∑60 i=1 e xi = 219.73. (e) Construct an approximate 95% confidence interval for ψ. You may find the following quantiles of the standard N (0, 1) distribution helpful: z0.025 = 1.96, z0.05 = 1.645, z0.1 = 1.28 where, for a given value ∈ (0, 1), z is the quantile such that P(Z > z) = for Z ∼ N (0, 1). [4] A Bayesian prior distribution for ψ is defined as pi(ψ) = βα Γ(α) 1 ψα+1 exp ( −β ψ ) . where α > 0 and β > 0 are known parameters. (f) Derive the posterior distribution of ψ and hence determine the Bayes’ estimate of ψ under quadratic error loss. [7] Continued STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 7 Q3. Suppose that X1, . . . , Xn are independent and identically distributed Bin(1, p) random variables. The parameter θ is such that θ = p2. (a) Find a minimal sufficient statistic for θ. [4] (b) Determine the maximum likelihood estimator of θ. Is this estimator unbiased for θ? Justify your answer. [7] The estimator U is defined U = { 1 if X1 = 1 and X2 = 1 0 otherwise (c) Show that U is unbiased for θ. [1] (d) Using the Rao-Blackwell theorem, determine an unbiased estimator of θ, denoting this estimator θ˜, such that Var(θ˜) ≤ Var(U). [7] (e) Is θ˜ the minimum variance unbiased estimator (MVUE) of θ? Justify your answer. [4] (f) Does the variance of θ˜ attain the Crame´r-Rao lower bound for the variance of unbi- ased estimators of θ? Justify your answer. [4] Turn Over STAT0008 Statistical Inference: Level 6 and 7 Examination 2020 Page 8 Q4. The random variables X1, . . . , Xn are independent and identically distributed, where Xi ∼ Exp(λ) for i ∈ {1, . . . , n} and λ > 0. (a) Based on the sample, X1, . . . Xn, construct the most powerful test of size α of the hypotheses H0 : λ = λ0 against H1 : λ = λ1 where α ∈ (0, 1), λ0 > 0, λ1 > 0 and λ1 > λ0. You may use the following results without proof: If Z1, . . . , Zn are independent Exp(θ) random variables then ∑n i=1 Zi ∼ Gamma(n, θ). If n ∈ N and Y ∼ Gamma(n, θ) then 2θY ∼ χ22n. If U ∼ χ2ν then P(U > χ2ν()) = for ∈ (0, 1). [9] (b) Based on the sample X1, . . . , Xn, construct a 100(1 − α)% confidence lower bound for λ. [3] (c) Derive the power function for the test that you constructed in (a). Your answer may be written in terms of a cumulative distribution function, F (.), where the explicit form of F need not be evaluated. [3] (d) Is the test that you constructed in (a) uniformly most powerful and of size α for a test of the hypotheses H0 : λ ≤ λ0 against H1 : λ > λ0? Justify your answer. [4] Associated with X1, . . . , Xn is the set of random variables {Z1, . . . , Zn} such that Zi = { 0 if Xi < γ; 1 if Xi ≥ γ; for i ∈ {1, . . . , n}, where γ > 0 is a known parameter. (e) Write down the likelihood function, L(λ, z), based on an observed sample z = (z1, . . . , zn) > of Z1, . . . , Zn and hence determine the maximum likelihood estimate of λ. [4] (f) State a condition that must be satisfied by X1, . . . , Xn to ensure that the maximum likelihood estimate of λ, derived in (e), is finite. [2] (g) Determine the Crame´r-Rao lower bound for the variance of unbiased estimators of λ, where λ is to be estimated using z1, . . . , zn. [4] End of Paper