# Emory and Henry College Compute the Values Statistics Questions

Problem 1: Efficient Exponential Expectation Esti-mators

Suppose that Yı, …, Y15 ~ Exp(\) are independent, with pdf

1

хе

exp ( – 6), 2 € (0,00)

Consider the following estimators:

5

în = X, Î2

=

=

ΣΧ;, = Υ,

i=1

Part A

Find ef f(, î2)

Part B

Find ef f(al, î3)

Part C

Find eff(A2, î3)

Part D

Do any of these estimators dominate another? If so, state which ones and explain

why.

Problem 2: Consistent Uniform Estimators

Suppose X1, …, Xn Unif(0,0) are independent, and suppose that n is an odd

number. Consider the following estimators:

n+1

6 = 2X, , X(n) , Ô3 = median(X1, …, Xn)

n

Part A

Which of these estimators are consistent? Give full justification.

Part B

Suppose that instead we are interested in estimating 02, using Ôz, ôz and Ôz. Which

of these estimators converge in probability to 02? Justify your answer.

Problem 3: Sufficient Estimators and the Likeli-

hood Function

In this problem we will explore the likelihood and determine whether or not estimators

are sufficient.

Part A

Suppose X1, …, Xn Pois() are independent. Find L (2X). Is X a sufficient

estimator for 1? Justify your answer using the factorization theorem.

Part B

Suppose X1, …, Xn – N(4,0%) are independent, and o2. is known. Find L (|X, 02).

Is X a sufficient estimator for ^? Justify your answer using the factorization theorem.

(Hint: look for a way to write X inside of the likelihood…)

Part C

Suppose that we have n independent realizations from a Unif(0,0) distribution. Is

X(n) a sufficient estimator for e? You do not need to use the factorization

theorem, but you must justify your answer.

0 =

Part D

Suppose that X1, …, Xn ~ Bern(p) are independent. Find L (P|X).

Part E

So far in this course we have been treating parameters as unknown constants, and

our data as random. Here we will take a look at what happens when we treat our

parameters as random and our data as fixed. Recall L (P|X) from above. If we treat

p as random and X as fixed, what distribution does p follow? Justify your answer.

(Hint: remember where p lives, your resulting distribution should be defined over the

р

same space).