Instructions: Answer each question on your own choice of paper

Q1. Let X₁, X₂, . . . , X_n be IID random variables with X_i ∼ Uniform(0, θ). Define a random variable

Y = max{X₁, X₂, . . . , X_n}.

It can be shown that, for any positive integer k,

E(Y^k) = n/(n+k) θ^k.

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 Var_θ(θ^{^}).

(iii) Suppose we are interested in estimating μ = E_θ(X). Let X^- be the sample average of {X_i : 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 MSE_θ(μ^{^}_c*) = σ²/(n + a). How does this compare with MSE_θ(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 {x_i : i = 1, 2, . . . , m}. For example, x_i is annual income for person i. Note that the x_i may have repeated values. (For example, x_i = 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 R_i ∼ Bernoulli(ρ) where 0 < ρ ≤ 1 is the sampling probability. We keep unit i in our sample if R_i = 1, otherwise i is not used. The R_i, i = 1, . . . ,m are independent draws.

(i) Define a random variable X_i = R_ix_i. What are the two possible values for X_i? What are E(X_i) and Var(X_i)?

(ii) Define the resulting sample size as N = R₁ + R₂ + · · · + R_m. 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 {x_i : i = 1, . . . ,m} that ensures m^-1 _i=1∑^mR_ix_i →^p ρμ as m → ∞, where μ = lim_m→∞(m^-1 _i=1∑^m x_i) is assumed to exist. [Hint: Use mean square convergence.]

(v) Argue that X^-_N = N^-1 _i=1∑^mR_ix_i is consistent for μ under the assumption in part (iv) - again, as the population size m → ∞.

Q4. Let {X_i : 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 h_g = n_g/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.