STATS 426 Winter, 2023 Exam #2
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STATS 426
Winter, 2023
Exam #2
PROBLEM # 1:Multiple Choice — Choose the correct answer. Each question below is worth (5) points.
1.1 Suppose X1 , · · · Xn are i.i.d. random variables (r.v.) with a common pdf f (x, θ) and we consider H0 : θ = θ0 , and HA : θ = θ 1 , (θ1 > θ0 ). Is the following statement correct? “If the test above is the most powerful (MP) test, then the test is the uniformly most powerful (UMP) for H0 : θ = θ0 , and HA : θ > θ0 .”
(A) Yes (B) No Explanation:
1.2 Consider X1 , X2 , . . . ,Xn are i.i.d. N (µ, σ2 ), Y1 , . . . ,Yn are i.i.d. N (µ, 4σ2 ), and Z1 , . . . ,Zn are i.i.d. N (µ, σ2 ); X , Y , Z , are the corresponding sample means. The X\ s, Y\ s and Z\ s are independent of each other. Then var(X + Y/2 + Z) is
(A) σ 2 (1 + .5 + 1) (B) σ 2 (1 + .5 + 1)/n (C) σ 2 (1 + .52 + 1) (D) σ 2 (1 + 1 + 1)/n (E) None of the above.
1.3 Following 1.2, we let S北(2), S2(2) , and Sy(2) be the corresponding sample variances. Which of the following statement is correct
(A) n3口21 S2(2) ~ χn(2)31 .
(B) n3口21 (S北(2) + S2(2)) ~ χ2(2)n31 .
(C) n3口21 (S北(2) + S2(2)/4) ~ χ2(2)n32 .
(D) none of above is true.
1.4 Following 1.1 and 1.2, we let S1(2) = (S北(2) + S2(2))/2, and S2(2) = (S北(2) + Sy(2))/2, and S3(2) = (S北(2) + S2(2) + Sy(2))/3, Please select the WRONG statement below.
(A) S1(2) is an unbiased estimator of σ 2 .
(B) S2(2) is an unbiased estimator of σ 2 .
(C) Z and S3(2) are independent.
(D) Two statements above are correct.
PROBLEM # 2: Suppose X1 , · · · Xn are i.i.d. r.v. with a common pdf f (x, β) = e3^α/8 , when
x > 0; and f (x, β) = 0, otherwise. β > 0.
2.1 (5 pt) Please provide the method of moment (MoM) estimator of β . What is the asymptotic variance of the MoM? If you cannot get the MoM, then obtain the asymptotic variance of ^X , where X is the sample mean of the X’s.
2.2 (5 pt) Please obtain a sufficient statistic, T, of β . Is the variance of your MoM estimator in 2.1 bigger than the Cram´er-Rao lower bound? Please justify your answer.
PROBLEM # 3: (5 pt) Suppose X1 , · · · X2n are i.i.d normal r.v.’s such that Xi ~ N (µ, 1), i = 1, · · · , 2n. Also, U1 = X1 , U2 = X2 , · · · , Un = Xn and V1 = Xn+1 , V2 = Xn+2 , · · · , Vn = X2n . We let U , V , X be the sample means of variables, U\ s, V\ s, and X\ s. Please use the Rao-Blackwell Theorem to show that var(X) < var(U). If you cannot use the Rao-Blackwell Theorem, you can use other ways to show var(X)
< var(U) and get maximum of 2 points.
PROBLEM # 4: (5 pt) Suppose X1 , · · · Xn are i.i.d. r.v with a common pdf f (x, β) = e3^北/8 , when
x > 0; and f (x, β) = 0, otherwise. β > 0. Please describe the testing procedure for the most powerful test with the significance level α when we test H0 : β = 2 vs Ha : β = β 1 (feel free to assume that n is large for your testing procedure). Here, β 1 > 2.
2023-04-14