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Department of Mathematics and StatisticsUniversity at Albany

Mathematical Statistics Preliminary ExamJune, 2017

Answer all questions.

1. Suppose X1, X2, and X3 form a random sample of size 3 from a distribution withprobability density function f(x; θ) = 1/θ, 0 < x < θ, where θ > 0.

a. Find a sufficient statistic for θ. Justify carefully!

b. Let Y1 < Y2 < Y3 denote the order statistics of this random sample. Consider thetest of H0 : θ = 1 vs H1 : θ < 1 given by rejecting H0 if y3 < 0.4 where y3 is theobserved value of Y3. Find the significance level α and the power function γ(θ),0 < θ < 1, of this test.

2. Recall the beta distribution with paramters α and β has probability density function

f(x) ={

Γ(α+β)Γ(α)Γ(β)

xα−1(1− x)β−1 if 0 < x < 10 elsewhere

where α > 0 and β > 0. Find the mean and variance of a random variable X with thisdistribution. Show all the steps.Note: A statement of the mean and variance without the proof of how they are calculatedwill not be considered.

3. Recall that the Γ(α, β) distribution has probability density function

f(x) ={

1Γ(α)βαx

α−1e−x/β if x > 00 elsewhere

where α > 0 and β > 0. Suppose the random variable X has this distribution.

a. Find the moment generating function of X.

b. Use the moment generating function of X to find E(X) and V ar(X).

c. If X1 has Γ(α1, β) distribution, X2 has Γ(α2, β) distribution, and X1 and X2 areindependent, find the distribution of X1 +X2. Justify!

4. Suppose Xi|θ ∼ i.i.d. Poisson(θ) (for i = 1, 2, ..., n), Θ ∼ Γ(α, β), α and β known.Find the posterior probability density function of Θ given X1 = x1, ..., Xn = xn, and findthe Bayes estimator of θ if the loss function is given by L(θ, δ(x)) = (θ − δ(x))2.

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5. Let {Xn} and X be random variables. Show that if Xn 7→ X in probability and g is acontinuous function, then g(Xn) 7→ g(X) in probability.

6. Let X1, · · · , Xn be independent identically distributed random variables that are uni-form on the interval [0, θ]. Let Yn = max(X1, · · · , Xn). Let Zn = n(θ − Yn). Show thatZn 7→ Z in distribution, where Z is an exponential random variable with mean θ.

7. Let X1, · · · , Xn be independent identically distributed random variables with commonprobability density function f(x|θ1, θ2) = 1

θ2exp (x− θ1)/θ2, x > θ1, −∞ < θ1 < ∞,

θ2 > 0. Find the MLE of θ1 and θ2.

8. Let X be a Poisson random variable with mean θ.

a. Determine the Fisher information I(θ), θ > 0.

b. Let H(X) = 1 if X = 0, and H(X) = 0 if X > 0. Calculate the informationinequality bound for H(X), that is, the Rao–Cramer lower bound. Show that thisbound is strictly less than the variance of H(X).

c. Show that H(X) is the minimum variance estimator of its expectation.

Note: For b & c, approach these problems thinking that you have a random sample ofsize n = 1, that is, we have a random sample X1 = X. For c, first explain why X1 is asufficient statistic and then show H(X1) is the minimum variance estimator.