The first correct answer gives 3 points; Any additional correct answer gives you 2 points, but any additional wrong answer

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

PROBLEM # 1:

Multiple Choice — Choose the correct answer. 

Each multiple choice question is worth (5) points.

The problems with * could have more than one answer.

The first correct answer gives 3 points; Any additional correct answer gives you 2 points, but any additional wrong answer will cost you 2 points.

1.1 ∗ Consider Y1, Y2, . . . ,Yn are i.i.d. random variables with n > 4 and T  is the sample mean

Y . Also let W  =  1 Σ4 Yi. Then

(A) E(Y1 + Y2|W ) = W 

(B) E(Y1 + W |W )  = E(Y3 + Y4|W )

(C)E(W |T ) = E(Y1|T )

(D) none of the above.

1.2   Let X and Y be 2 random variables: X ∼ Exp(λ = 1) and Y ∼ Exp(λ = 0.5). Then we know that Cov(X, Y ) is (A) ≥ −1 (B) ≥ 0 (C) ≤ 1 (D) ≤ 2 (E) None of the above is correct.

1.3 ∗ Let X be a continuous random variable with pdf, fX(x), and fX(t + 3) = fX (3 − t) for all

t > 0. Which of the following statement is correct? (A) P (X > 3) = 0.5 (B) P (X ≤ 3) > P (X ≥ 3) (C) E(X) = 3 (D) ∫3∞ fX(x)dx < P (X < 3).

1.4 ∗ Let X1, X2, . . . ,Xn be i.i.d random variables with the density fX(x; θ). Which of the following statements are correct?

(A) If θˆ is an unbiased estimator of θ, then 2θˆ is unbiased for 2θ.

(B) If θˆ is an unbiased estimator of θ, then θˆ2  is unbiased for θ2.

(C) If θˆ is a consistent estimator of θ, then θˆ2  is a consistent estimator of θ2.

(D) If T = (T1, T2) provides a sufficient statistic for θ, then W = (T1 +T2, T1 − T2) also provides a sufficient statistic for θ.

PROBLEM # 2: Suppose X1, X2, . . . ,Xn are i.i.d Exponential(θ), Y1, Y2, . . . ,Yn are i.i.d Exponential(3θ), and they are mutually independent.

2.1 Find a sufficient statistic that consists of all X and Y observations.

2.2 Use observations from both X and Y to obtain the MLE of θ.

2.3 Please provide a 95% confidence interval for θ and justify your answer (feel free to assume that n is large). 

PROBLEM # 3: Let X1, X2, X3 follow a multinomial distribution with parameters n and π = (π1, π2, π3).

3.1 Consider the scenario that π1 = π2 = 1 − θ, and π3 = θ, please describe the testing procedure for the most powerful test with the significance level α when we test H0 : θ = 0.2 vs Ha : θ = 0.4 (feel free to assume that n is large for your testing procedure).

3.2 Now to check the assumed structure, we let the H0 be π1 = π2  = 1 − θ, and π3  = θ, and Ha be that the H0 is incorrect. Please provide a test statistic and the corresponding testing procedure for a generalized likelihood ratio test with the significance level α for this problem. Feel free to use the notation of χ2(α) as

the 1 − α quantile of the Chi-square distribution with the degree of freedom q. 

PROBLEM # 4: Suppose we observe 3 sets of independent random variables:  X1,  X2,  . . . ,Xn ∼

N(θx,  σ2); Y1, Y2, . . . ,Yn ∼ N(θy,  σ2); and Z1, Z2, . . . ,Zn ∼ N(θx + θy,  σ2), which are also independent of

each other.  Let S2 = Σn   (Xi − X)2/(n−1), S2 = Σn   (Yi − Y )2/(n−1),and S2 = Σn   (Zi − Z)2/(n−1),

where X, Y  and Z are respectively the sample means for the X’s, Y ’s and Z’s. Also  we  let  S2  =

(1/3)(S2 + S2 + S2).

x y z

4.1 Use all observations to obtain the MLE of σ2.

4.2 What  is  the  asymptotic  distribution  of  the  MLE,  σˆ2 

? If you have a problem to find the answer, 

find the asymptotic distribution of S2 = (1/3)(S2 + S2 + S2) to obtain maximum of 4 points instead (you

x y z

do not need to do this if you obtain the answer for the MLE).

4.3 Let

X + Y − Z

T = .S

One argues that a × T, where a is a constant, follows a t-distribution. If you agree with the statement, please provide the constant a and an appropriate degree of freedom, and justify this statement. Provide justification why this is incorrect if you don’t agree with the statement.

Attachments:

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