STAT 450: Final Examination Version 1. Richard Lockhart 16 December 2002

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1 Name: Last Name 1, First Name 1 Stdnt # StudentNumber1 STAT 450: Final Examination Version 1 Richard Lockhart 16 December 2002 Instructions: This is an open book exam. You may use notes, books and a calculator. You may not use calculators capable of remote communication nor may you use cellular phones. The exam is out of 50. There are 14 pages including this one and two pages of extra working space. Question Maximum Mark Question Maximum Mark 1a 5 1b 5 2a 4 2b 1 3a 2 3b 2 3c 2 3d 3 4a 1 4b 1 4c 5 4d 5 4e 3 4f 1 5a 3 5b 1 5c 1 5d 5 Total Out of 50 1

2 1. Suppose that X 1,...,X n are independent random variables. Suppose that the density X k is { } 1 f k (x;σ) = 2k2 πσ exp x2 2k 2 σ 2 where the unknown parameter σ belongs to (0, ). (a) Find the UMVUE of σ 2. (5 marks) 2

3 (b) Find the Cramér-Rao Lower Bound for the variance of unbiased estimators of σ 4. (5 marks) 3

4 2. Suppose X 1,...,X n are a sample from the density 2φ(x 2) 2φ 1 2 x 3 f(x;φ) = 0 otherwise (a) Find the moment generating function of 6log(X 1 2). (4 marks) (b) What is the distribution of 6 n i=1 log(x i 2)? (1 mark) 4

5 3. Suppose X 1,...,X n are iid random variables with (1 θ) (1+x)/2 θ (1 x)/2 x = 1, 1 P(X = x;θ) = 0 otherwise for an unknown θ [0,1]. (a) Find constants a, b, c, and d such that T = (ax 1 +b)(cx 2 +d) is an unbiased estimator of φ (2θ 1) 2. (2 marks) (b) Find the variance of the estimator T above. (2 marks) 5

6 (c) Find constants a, b and c so that T = a X2 +b X +c is an unbiased estimate of φ. (2 marks) (d) Which estimate, T or T has smaller variance. Explain clearly how you know. (3 marks) 6

7 4. Suppose that X 1,X 2,X 3 are 3 random variables. Assume i) Given X 1 = x 1 and X 2 = x 2 the random variable X 3 has a normal distribution with mean 2ρx 2 and variance σ 2, ii) Given X 1 = x 1 the random variable X 2 has a normal distribution with mean 2ρx 1 and variance σ 2. iii) The random variable X 1 has a N(0,σ 2 ) distribution. (a) What is the joint density of X 1 and X 2? (1 mark) (b) What is the joint density of X 1, X 2 and X 3? (1 mark) 7

8 (c) Show that X 1, X 2 2ρX 1 and X 3 2ρX 2 are independent N(0,σ 2 ) random variables. NOTE: you may use this fact in the rest of the problem even if you could not do this part. (3 marks) (d) What is the distribution of X 2 1 +(X 2 2ρX 1 ) 2 σ 2 (1 mark) 8

9 (e) Find the MLE of (ρ,σ) if we observe X 1,X 2,X 3. (5 marks) 9

10 (f) Find the Fisher information matrix for this problem. (5 marks) 10

11 5. Suppose X, Y have joint density φ e x xy > 1,x > 0,y > 0 x f(x,y;φ) = φ y φ+1 0 otherwise for some unknown φ > 0. (a) Suppose we use the rejection region R = {(x,y) : x > log(4)} What is the power function of this rejection region. (3 marks) (b) If the null hypothesis is φ 1 what is the level (or size) of the test? (1 mark) (c) What is the probability of a type II error if φ = 2? (1 mark) 11

12 (d) Find the Uniformly Most Powerful test at level α of φ = 1 against φ > 1. Express the rejection region in the form S(X 1,...,X n ) < k for some statistic S and give an explicit formula for k. (5 marks) 12

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15 Name: Last Name2, First Name 2 Stdnt # StudentNumber1 STAT 450: Final Examination Version 2 Richard Lockhart 16 December 2002 Instructions: This is an open book exam. You may use notes, books and a calculator. You may not use calculators capable of remote communication nor may you use cellular phones. The exam is out of 50. There are 14 pages including this one and two pages of extra working space. Question Maximum Mark Question Maximum Mark 1a 5 1b 5 2a 4 2b 1 3a 2 3b 2 3c 2 3d 3 4a 1 4b 1 4c 5 4d 5 4e 3 4f 1 5a 3 5b 1 5c 1 5d 5 Total Out of 50 1

16 1. Suppose that X 1,...,X n are independent random variables. Suppose that the density X k is { } 1 f k (x;σ) = 2k3 πσ exp x2 2k 3 σ 2 where the unknown parameter σ belongs to (0, ). (a) Find the UMVUE of σ 2. (5 marks) 2

17 (b) Find the Cramér-Rao Lower Bound for the variance of unbiased estimators of σ 3. (5 marks) 3

18 2. Suppose X 1,...,X n are a sample from the density 3θ(x 4) 3θ 1 4 x 5 f(x;θ) = 0 otherwise (a) Find the moment generating function of 5log(X 1 4). (4 marks) (b) What is the distribution of 5 n i=1 log(x i 4)? (1 mark) 4

19 3. Suppose X 1,...,X n are iid random variables with ψ (1+x)/2 (1 ψ) (1 x)/2 x = 1, 1 P(X = x;ψ) = 0 otherwise for an unknown ψ [0,1]. (a) Find constants r, s, t, and u such that S = (rx 1 +s)(tx 2 +u) is an unbiased estimator of γ (1 2ψ) 2. (2 marks) (b) Find the variance of the estimator S above. (2 marks) 5

20 (c) Find constants r, s and t so that S = r X2 +s X +t is an unbiased estimate of γ. (2 marks) (d) Which estimate, S or S has smaller variance. Explain clearly how you know. (3 marks) 6

21 4. Suppose that X 1,X 2,X 3 are 3 random variables. Assume i) Given X 1 = x 1 and X 2 = x 2 the random variable X 3 has a normal distribution with mean 3ρx 2 and variance σ 2, ii) Given X 1 = x 1 the random variable X 2 has a normal distribution with mean 3ρx 1 and variance σ 2. iii) The random variable X 1 has a N(0,σ 2 ) distribution. (a) What is the joint density of X 1 and X 2? (1 mark) (b) What is the joint density of X 1, X 2 and X 3? (1 mark) 7

22 (c) Show that X 1, X 2 3ρX 1 and X 3 3ρX 2 are independent N(0,σ 2 ) random variables. NOTE: you may use this fact in the rest of the problem even if you could not do this part. (3 marks) (d) What is the distribution of X 2 1 +(X 2 3ρX 1 ) 2 σ 2 (1 mark) 8

23 (e) Find the MLE of (ρ,σ) if we observe X 1,X 2,X 3. (5 marks) 9

24 (f) Find the Fisher information matrix for this problem. (5 marks) 10

25 5. Suppose X, Y have joint density φ e x xy > 1,x > 0,y > 0 x f(x,y;φ) = φ y φ+1 0 otherwise for some unknown φ > 0. (a) Suppose we use the rejection region R = {(x,y) : x > log(5)} What is the power function of this rejection region. (3 marks) (b) If the null hypothesis is φ 1 what is the level (or size) of the test? (1 mark) (c) What is the probability of a type II error if φ = 2? (1 mark) 11

26 (d) Find the Uniformly Most Powerful test at level α of φ = 1 against φ > 1. Express the rejection region in the form S(X 1,...,X n ) < k for some statistic S and give an explicit formula for k. (5 marks) 12

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29 Name: Last Name 3, First Name 3 Stdnt # StudentNumber1 STAT 450: Final Examination Version 3 Richard Lockhart 16 December 2002 Instructions: This is an open book exam. You may use notes, books and a calculator. You may not use calculators capable of remote communication nor may you use cellular phones. The exam is out of 50. There are 14 pages including this one and two pages of extra working space. Question Maximum Mark Question Maximum Mark 1a 5 1b 5 2a 4 2b 1 3a 2 3b 2 3c 2 3d 3 4a 1 4b 1 4c 5 4d 5 4e 3 4f 1 5a 3 5b 1 5c 1 5d 5 Total Out of 50 1

30 1. Suppose that X 1,...,X n are independent random variables. Suppose that the density X k is { } 1 f k (x;τ) = 2k4 πτ exp x2 2k 4 τ 2 where the unknown parameter τ belongs to (0, ). (a) Find the UMVUE of τ 2. (5 marks) 2

31 (b) Find the Cramér-Rao Lower Bound for the variance of unbiased estimators of τ 5. (5 marks) 3

32 2. Suppose X 1,...,X n are a sample from the density 3θ(x 6) 3θ 1 6 x 7 f(x;θ) = 0 otherwise (a) Find the moment generating function of 3log(X 1 6). (4 marks) (b) What is the distribution of 3 n i=1 log(x i 6)? (1 mark) 4

33 3. Suppose X 1,...,X n are iid random variables with (1 γ) (1+x)/2 γ (1 x)/2 x = 1, 1 P(X = x;γ) = 0 otherwise for an unknown γ [0,1]. (a) Find constants a, b, c, and d such that U = (ax 3 +c)(bx 4 +d) is an unbiased estimator of φ (2γ 1) 2. (2 marks) (b) Find the variance of the estimator U above. (2 marks) 5

34 (c) Find constants a, b and c so that U = a +b X2 +c X is an unbiased estimate of φ. (2 marks) (d) Which estimate, U or U has smaller variance. Explain clearly how you know. (3 marks) 6

35 4. Suppose that X 1,X 2,X 3 are 3 random variables. Assume i) Given X 1 = x 1 and X 2 = x 2 the random variable X 3 has a normal distribution with mean 4ρx 2 and variance τ 2, ii) Given X 1 = x 1 the random variable X 2 has a normal distribution with mean 4ρx 1 and variance τ 2. iii) The random variable X 1 has a N(0,τ 2 ) distribution. (a) What is the joint density of X 1 and X 2? (1 mark) (b) What is the joint density of X 1, X 2 and X 3? (1 mark) 7

36 (c) Show that X 1, X 2 4ρX 1 and X 3 4ρX 2 are independent N(0,τ 2 ) random variables. NOTE: you may use this fact in the rest of the problem even if you could not do this part. (3 marks) (d) What is the distribution of X 2 1 +(X 2 4ρX 1 ) 2 τ 2 (1 mark) 8

37 (e) Find the MLE of (ρ,τ) if we observe X 1,X 2,X 3. (5 marks) 9

38 (f) Find the Fisher information matrix for this problem. (5 marks) 10

39 5. Suppose X, Y have joint density φ e x xy > 1,x > 0,y > 0 x f(x,y;φ) = φ y φ+1 0 otherwise for some unknown φ > 0. (a) Suppose we use the rejection region R = {(x,y) : x > log(6)} What is the power function of this rejection region. (3 marks) (b) If the null hypothesis is φ 1 what is the level (or size) of the test? (1 mark) (c) What is the probability of a type II error if φ = 2? (1 mark) 11

40 (d) Find the Uniformly Most Powerful test at level α of φ = 1 against φ > 1. Express the rejection region in the form S(X 1,...,X n ) < k for some statistic S and give an explicit formula for k. (5 marks) 12

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43 Name: Last Name 4, First Name4 Stdnt # StudentNumber1 STAT 450: Final Examination Version 4 Richard Lockhart 16 December 2002 Instructions: This is an open book exam. You may use notes, books and a calculator. You may not use calculators capable of remote communication nor may you use cellular phones. The exam is out of 50. There are 14 pages including this one and two pages of extra working space. Question Maximum Mark Question Maximum Mark 1a 5 1b 5 2a 4 2b 1 3a 2 3b 2 3c 2 3d 3 4a 1 4b 1 4c 5 4d 5 4e 3 4f 1 5a 3 5b 1 5c 1 5d 5 Total Out of 50 1

44 1. Suppose that X 1,...,X n are independent random variables. Suppose that the density X k is { } 1 f k (x;τ) = 2k5 πτ exp x2 2k 5 τ 2 where the unknown parameter τ belongs to (0, ). (a) Find the UMVUE of τ 2. (5 marks) 2

45 (b) Find the Cramér-Rao Lower Bound for the variance of unbiased estimators of τ 6. (5 marks) 3

46 2. Suppose X 1,...,X n are a sample from the density 3θ(x 8) 3θ 1 8 x 9 f(x;θ) = 0 otherwise (a) Find the moment generating function of 4log(X 1 8). (4 marks) (b) What is the distribution of 4 n i=1 log(x i 8)? (1 mark) 4

47 3. Suppose X 1,...,X n are iid random variables with (1 δ) (1+x)/2 δ (1 x)/2 x = 1, 1 P(X = x;δ) = 0 otherwise for an unknown δ [0,1]. (a) Find constants a, b, c, and d such that V = (ax 3 +c)(dx 4 +b) is an unbiased estimator of ψ (2δ 1) 2. (2 marks) (b) Find the variance of the estimator V above. (2 marks) 5

48 (c) Find constants a, b and c so that V = a +b X +c X2 is an unbiased estimate of ψ. (2 marks) (d) Which estimate, V or V has smaller variance. Explain clearly how you know. (3 marks) 6

49 4. Suppose that X 1,X 2,X 3 are 3 random variables. Assume i) Given X 1 = x 1 and X 2 = x 2 the random variable X 3 has a normal distribution with mean 5ρx 2 and variance τ 2, ii) Given X 1 = x 1 the random variable X 2 has a normal distribution with mean 5ρx 1 and variance τ 2. iii) The random variable X 1 has a N(0,τ 2 ) distribution. (a) What is the joint density of X 1 and X 2? (1 mark) (b) What is the joint density of X 1, X 2 and X 3? (1 mark) 7

50 (c) Show that X 1, X 2 5ρX 1 and X 3 5ρX 2 are independent N(0,τ 2 ) random variables. NOTE: you may use this fact in the rest of the problem even if you could not do this part. (3 marks) (d) What is the distribution of X 2 1 +(X 2 5ρX 1 ) 2 τ 2 (1 mark) 8

51 (e) Find the MLE of (ρ,τ) if we observe X 1,X 2,X 3. (5 marks) 9

52 (f) Find the Fisher information matrix for this problem. (5 marks) 10

53 5. Suppose X, Y have joint density φ e x xy > 1,x > 0,y > 0 x f(x,y;φ) = φ y φ+1 0 otherwise for some unknown φ > 0. (a) Suppose we use the rejection region R = {(x,y) : x > log(7)} What is the power function of this rejection region. (3 marks) (b) If the null hypothesis is φ 1 what is the level (or size) of the test? (1 mark) (c) What is the probability of a type II error if φ = 2? (1 mark) 11

54 (d) Find the Uniformly Most Powerful test at level α of φ = 1 against φ > 1. Express the rejection region in the form S(X 1,...,X n ) < k for some statistic S and give an explicit formula for k. (5 marks) 12

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