Reference no: EM131746396
Instructions: Answer each question on your own choice of paper
Q1. Let X1, X2, . . . , Xn be IID random variables with Xi ∼ Uniform(0, θ). Define a random variable
Y = max{X1, X2, . . . , Xn}.
It can be shown that, for any positive integer k,
E(Yk) = n/(n+k) θk.
You need not prove this (although it is a good exercise).
(i) Use the expression above with k = 1 to propose an unbiased estimator of θ based on Y. Call the estimator θ^. Does the estimator make intuitive sense?
(ii) Find Varθ(θ^).
(iii) Suppose we are interested in estimating μ = Eθ(X). Let X- be the sample average of {Xi : i = 1, . . . ,n}. Find Var θ(X-).
(iv) Obtain an unbiased estimator of μ using θ^ from part (i). Call the estimator μ^.
(v) Do you prefer X- or μ^ for estimating μ? Explain.
(vi) How come your finding in part (v) does not contradict the result that X- BLUE for μ for any distribution with finite second moment?
Q2. For X ≥ 0 with E(X2) < ∞ and E(X) = μ, define a class of estimators of μ as μ^c,m = cX- + (1 - c)m,
where c and M are nonnegative constants.
(i) Show that μ^c,m can be written
μ^c,m = X- - (1 - c)(X- - m).
If 0 < c < 1, argue that μ^c,m "shrinks" X- toward m in the following sense: If X- > m then m < μ^c,m < X-; if X- < m then m > μ^c,m > X-.
(ii) Find the bias in μ^c,m.
(iii) Find Var(μ^c,m). What is its MSE(μ^c,m)?
(iv) Now take m = 0, and consider the estimators μ^c = cX- we studied in lecture. Suppose that a = σ2/μ2 is known, so we can choose
c* = n/(n + a)
to minimize the MSE. Show that MSEθ(μ^c*) = σ2/(n + a). How does this compare with MSEθ(X)?
Q3. Consider a setting with a finite population where there are m units in the population. List the outcomes for the population units as {xi : i = 1, 2, . . . , m}. For example, xi is annual income for person i. Note that the xi may have repeated values. (For example, xi = 1 if person i has a college degree, zero otherwise. There will be many people in the population in each category.)
Now we draw a sample from the population using the following sampling scheme. For each i, generate a random variable Ri ∼ Bernoulli(ρ) where 0 < ρ ≤ 1 is the sampling probability. We keep unit i in our sample if Ri = 1, otherwise i is not used. The Ri, i = 1, . . . ,m are independent draws.
(i) Define a random variable Xi = Rixi. What are the two possible values for Xi? What are E(Xi) and Var(Xi)?
(ii) Define the resulting sample size as N = R1 + R2 + · · · + Rm. What is the distribution of N? What is E(N/m)?
(iii) Now consider what happens as the population size grows, so m → ∞. Argue that N/m→p ρ.
(iv) Impose a condition on the {xi : i = 1, . . . ,m} that ensures m-1 i=1∑mRixi →p ρμ as m → ∞, where μ = limm→∞(m-1 i=1∑m xi) is assumed to exist. [Hint: Use mean square convergence.]
(v) Argue that X-N = N-1 i=1∑mRixi is consistent for μ under the assumption in part (iv) - again, as the population size m → ∞.
Q4. Let {Xi : i = 1, 2, . . . } be IID random variables from the Pareto(α, β) distribution with CDF
F(x; α, β) = 1- (1 + x/β)-α, x > 0
= 0, x ≤ 0
for parameters α, β > 0. The mean and variance are given in the lecture notes on distributions. Let X-n be the sample average of the first n draws.
(i) What condition is needed on α to ensure √n(X-n - μ) is asymptotically normal? Find AVar[√n(X-n - μ)].
(ii) Suppose α = 3 and want to estimate γ = P(X > 2). Find γ as a function of μ and propose a consistent estimator of γ using X-n. Call it γ^n.
(iii) Find AVar[√n(γ^n - γ)], where γ^n is the estimator in part (iii). (Note: The algebra is a bit messy.)
Q5. Consider independent sampling from two populations indexed by their means and variances: (μ1, σ12) and (μ2, σ22). You obtain a random sample from each population, of sizes n1 and n2, respectively. Therefore, sampling is independent both within and across populations. Let X-1 and X-2 denote the sample averages, and let n = n1 + n2 be the total sample size.
(i) Define γ = μ1 - μ2. Argue that γ^ = X-1 - X-2 is unbiased for γ. Show that Var(γ^) can be written as
Var(γ^) = n-1(σ12/h1 + σ22/h2),
where hg = ng/n, g = 1, 2 are the fractions of observations in each population.
(ii) Assume that as n →∞, n1/n → ρ1 > 0 and n2/n → ρ2 > 0, where ρ1 + ρ2 = 1. Show that
AVar[√n(γ^ - γ) = σ12/ρ1 + σ22/ρ2.