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2014.10.24

1. (15%) Suppose we toss a coin N times independently and let p be the probability of getting head when N = 1. Let X and Y denote the number of heads and the number of tails respectively.

(a) (5%) Prove that X andY are dependent when N = 1.

{b) (10%) Suppose that the coin is tossed N times. The number of tosses N is a random variable and N"' Poisson(>.). Again, denote the number of heads and tails by X and Y, respectively. Show that X and Y are independent.

2. (20%) Let X1 , · · ·, Xn "' Uniform(O, 1) and Yn

=

X~. Here Xn is the average of

X ^{1 , · ·}^·^,X n. Find the asymptotic distribution of Yn. (i.e., Find an and b such that

an (Yn - b) converges to a non-degenerate distribution.)

3. (20%) Let X1 , X2 , ... be a sequence of independent and identically distributed random variables with density!(·). Suppose that P(Xi

>

0) = 1 and that)..= limx~o f(x)

>

0. Set X(1) to be min{X1 , · · · ,Xn} and Yn = nX(1). Determine the asymptotic distribution of Yn.

4. (20%) Consider the one-sample problem:

Yi

"'N(J-L, 1), 1 ::;

i::;

n with the }'is i.i.d. (a) (5%) Determine ftc which is the maximum likelihood estimator of J-L when IJ-£1²^::;c.

Here c ~ 0.

(b) (7%) Determine the mean square error of ftc· (If you are not sure on your answer obtained in (a), you can assume that ftc is

Y

/(1

+

c/n) where

Y

is the average of }'i, 1::;

i::;

n.)

(c) (8%) Determine ftLasso which is the maximum likelihood estimator of J-L when IJ-LI ::; c. Here c ~ 0.

5. (25%) Let X1, X2 , ... , Xn be independent and identically distributed normally distributed random variables with mean

e

and variance 1. Consider testing H_{0 :}

e

^{= 0}

versus Ha : e =en. Here en

>

0.

(a) (7%) Determine the rejection region of the most powerful test at level a, 0 <

a < 1. Give reason to justify your answer.

(b) (8%) Find the power of the test you have in (a) under Ha when en = 1/fo.

(i.e. Fnd j3(en).) If you are not sure that your answer of (a) is correct, you can answer (b) by assuming that the rejection region is R = {n-¹2:::~=1 ^Xi

> en} ·

You then need to determine en-

(c) (10%) Determine the limit of j3(en) with en= 1/fo as n goes to infinity.