Homework Assignments

1 HW11

2 Homework Assignment 2Q 2.1 Consider a random sample of size n from a double exponential distribution,X i ∼ DE(θ, η).(a) Find the MLE of η, when θ = 1. (Comment. at some stage you mightcome across median as an estimator.)(b) Find the MLE’s when both θ and η are unknown.Q 2.2 Consider a random sample of size n from a uniform distribution, X ∼ UNIF (−θ, θ);θ > 0. Find a constant c such that c(X n:n − X 1:n ) is an unbiased estimator ofθ.Q 2.3 Let S be the sample standard deviation, based on a random sample of sien from a distribution with pdf f(x; µ, σ 2 ) with mean µ, and variance σ 2 .(a) Show that E(S) ≤ σ, where equality holds iff f(x; µ, σ 2 ) is degenerate atµ, i.e., P [X = µ] = 1. (Hint. Consider Var(S))(b) If X i ∼ N(µ, σ 2 ), find a constant c such that cS is an unbiased estimatorof σ. (Hint. Prove (n − 1)S 2 /σ 2 ∼ χ 2 (n − 1), and S = √ S 2 .)Q 2.4 Consider a random sample of size n from a Bernoulli distribution, X i ∼BIN(1, p). For a uniform prior density, p ∼ UNIF (0, 1), and squared errorloss, find the following:(a) Bayes estimator of p.(b) Bayes estimator of p(1 − p).(c) Bayes risk for the estimator in (a).Q 2.5 Consider a random sample of size n from a distribution with discrete pdf(f(x; p) = p(1 − p) x , x = 0, 1, . . ., and zero otherwise.(a) Find the MLE of p.(b) Find the MLE of θ = (1 − p)/p.(c)Find the CRLB for variances of unbiased estimators of θ.(d) Is the MLE of θ a UMVUE?(e) Is the MLE of θ MSE consistent?(f) Find the asymptotic distribution of the MLE of θ.(g) Let ˜θ = n ¯X/(n + 1). Find risk functions of both ˜θ and ¯X using the lossfunction L(t; θ) = (t − θ) 2 /(θ 2 + θ).Q 2.6 Let X be a random variable with CDF F (x).(a) Show that E[(X − c) 2 ] is minimized by the value c = E(X).(b) Assuming that X is continuous, show that E[|X − c|] is minimized if cis the median, that is, the value such that F (c) = 1/2.2

3 Homework Assignment 3Q 3.1 Consider a random sample of size n from a two-parameter exponentialdistribution X i ∼ EXP (1, η).(a) Show that S = X 1:n is sufficient for η by showing that the conditionaldensity of the joint pdf given S is free of η.(b) Show that S also is complete.(c) Verify that X 1:n − (1/n) is the UMVUE of η.(d) Find the UMVUE of the pth percentile.Q 3.2 Let X ∼ N(0, θ); θ > 0.(a) Show that X 2 is complete and sufficient for θ.(b) Show that N(0, θ) is not a complete family.Q 3.3 Show that N(µ, µ 2 ) does not belong to the REC.Q 3.4 Consider a random sample of size n from UNIF (θ, 2θ), θ > 0. Can youfind a single sufficient statistic for θ? Can you find a pair of jointly sufficientstatistics for θ?Q 3.5 Let X 1 , . . . , X n be a random sample from a Bernoulli distribution, X i ∼BIN(1, p), 0 < p < 1.(a) Find the UMVUE of V ar(X) = p(1 − p).(b) Find the UMVUE of p 2 .Q 3.6 Consider a random sample of size n from a Poisson distribution, X i ∼P OI(µ), µ > 0.(a) Find the UMVUE of P [X = 0] = e −µ .(b) Show that S = (−1) X is the UMVUE of e −2µ .Q 3.7 Consider a random sample of size n from a gamma distribution, X i ∼GAM(θ, κ), and let ¯X = (1/n)∑Xi and ˜X = ( ∏ X i ) 1/n be the sample meanand geometric mean, respectively.(a) Show that ¯X and ˜X are jointly complete and sufficient for θ and κ.(b) Find the UMVUE of µ = θκ.(c) Find the UMVUE of µ n .(d) Show that the distribution of T = ¯X/ ˜X does not depend on θ.(e) Show that ¯X and T are stochastically independent random variables.(f) Show that the conditional pdf of ¯X given ˜X = ˜x does not depend on κ.Q 3.8 Consider a random sample of size n from a two-parameter exponentialdistribution X i ∼ EXP (θ, η).(a) Show that X 1:n and T = X 1:n − ¯X are stochastically independent.(b) Find the MLE ˆθ of θ.(c) Find the UMVUE of η.(d) Show that the conditional pdf of X 1:n given ¯X does not depend on θ.(e) Show that the distribution of Q = (X 1:n − η)/ˆθ is free of η, θ.3