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Question 1 (*** — Pareto distribution (50%)). Let X1,..., Xnfo, where the PDF fo is given...
Let X1,... , Xn be a random sample from the Pareto distribution with pdf Ox (0+1), x > 1, f(z0) where 0>0 is unkn...
Let X1,... , Xn be a random sample from the Pareto distribution with pdf Ox (0+1), x > 1, f(z0) where 0>0 is unknown (a) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = T log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval (b) Find a confidence interval for 0 with confidence coefficient 1 - a by pivoting the cdf...
Let X1,... , Xn be a random sample from the Pareto distribution with pdf { f (r0)= 0, where 0>0 is unknown (a) Find...
Let X1,... , Xn be a random sample from the Pareto distribution with pdf { f (r0)= 0, where 0>0 is unknown (a) Find a uniformly most powerful (UMP) test of size a for testing Ho 0< 0 versus where 0o>0 is a fixed real number. (Use quantiles of chi-square distributions to express the test) (b) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = log Xi. (Use quantiles...
Let X1, ..., X, bei.i.d. random variable with pdf fe defined as follows: fo (2) =...
Let X1, ..., X, bei.i.d. random variable with pdf fe defined as follows: fo (2) = 0x0-11(0<x< 1) where 0 is some positive number. Using the MLE Ô, find the shortest confidence interval for 6 with asymptotic level 85% using the plug-in method. To avoid double jeopardy, you may use V for the appropriate estimator of the asymptotic variance V (Ô), and/or I for the Fisher information I (Ô) evaluated at ê, or you may enter your answer without using...
Question 2 Let X Pareto(r, 8 = 1) which has pdf: f(x) = 1 , 1...
Question 2 Let X Pareto(r, 8 = 1) which has pdf: f(x) = 1 , 1 >1 and r > 1 (a) Given a random sample of size n from X show that the mle for r is: r* = 1/7 where Y = SEY and Y = log X (b) Let Y = log X Use the mgf technique (with t <r) to show that: Y Exp(1 = r) [ HINT: My(t) = Eletbox] = E[X“) = * **f(x)dt...
6. Suppose that the statistical model is given by fo(1) fo(2) fo(3) fol(4) θ=a11/3 1/6 1/3 1/6 0-...
6. Suppose that the statistical model is given by fo(1) fo(2) fo(3) fol(4) θ=a11/3 1/6 1/3 1/6 0-b 1/21/4 1/8 1/8 and that the priors is given by π(a) = 1/3,π(b) = 2/3 and we observe the sample (x1両,xs) = (1, i, 3). (a) (5 marks) Determine the posterior of (c) (5 marks) Determine the MAP estimate of θ (d) (5 marks) Determine the relative belief estimate of θ
6. Suppose that the statistical model is given by fo(1) fo(2)...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y)...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
4.(120) Let X1,,,Xn be iid r(, 1) and g(u) given. Let 6n be the MLE of g(4) (1)(60) Find the asymptotic distributio...
4.(120) Let X1,,,Xn be iid r(, 1) and g(u) given. Let 6n be the MLE of g(4) (1)(60) Find the asymptotic distribution of 6, (2)(60) Find the ARE of T Icc(X) w.r.t. on P(X1> c), c > 0 is i n i1 5.(80) Let X1, ,,Xn be iid with E(X1) = u and Var(X1) limiting distribution of nlog (1 +). o2. Find the where T n(X - 4)/s. - 1 -
4.(120) Let X1,,,Xn be iid r(, 1) and g(u)...
have the prior pdf 2-2) Let a (e)-0<a<e where B>1 1 Find the joint pdf fo,...
have the prior pdf 2-2) Let a (e)-0<a<e where B>1 1 Find the joint pdf fo, e) of Y and e Use f(y.0) py0)2(0) [Hint]
have the prior pdf 2-2) Let a (e)-0
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1...
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1 +0)-r(0-1) (1-2), 0 < x < 1, θ > 0. the estimator T-4 is a method of moments estimator for θ. It can be shown that the asymptotic distribution of T is Normal with ETT θ and Var(T) 0042)2 Apply the integral transform method (provide an equation that should be solved to obtain random observations from the distribution) to generate a sam ple of...
Let X1,.. ,X be a random sample from an N(p,02) distribution, where both and o are...
Let X1,.. ,X be a random sample from an N(p,02) distribution, where both and o are unknown. You will use the following facts for this ques- tion: Fact 1: The N(u,) pdf is J(rp. σ)- exp Fact 2 If X,x, is a random sample from a distribution with pdf of the form I-8, f( 0,0) = for specified fo, then we call and 82 > 0 location-scale parameters and (6,-0)/ is a pivotal quantity for 8, where 6, and ô,...
Let X1,... , Xn be a random sample from the Pareto distribution with pdf Ox (0+1), x > 1, f(z0) where 0>0 is unknown (a) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = T log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval (b) Find a confidence interval for 0 with confidence coefficient 1 - a by pivoting the cdf...
Let X1,... , Xn be a random sample from the Pareto distribution with pdf { f (r0)= 0, where 0>0 is unknown (a) Find a uniformly most powerful (UMP) test of size a for testing Ho 0< 0 versus where 0o>0 is a fixed real number. (Use quantiles of chi-square distributions to express the test) (b) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = log Xi. (Use quantiles...
Let X1, ..., X, bei.i.d. random variable with pdf fe defined as follows: fo (2) = 0x0-11(0<x< 1) where 0 is some positive number. Using the MLE Ô, find the shortest confidence interval for 6 with asymptotic level 85% using the plug-in method. To avoid double jeopardy, you may use V for the appropriate estimator of the asymptotic variance V (Ô), and/or I for the Fisher information I (Ô) evaluated at ê, or you may enter your answer without using...
Question 2 Let X Pareto(r, 8 = 1) which has pdf: f(x) = 1 , 1 >1 and r > 1 (a) Given a random sample of size n from X show that the mle for r is: r* = 1/7 where Y = SEY and Y = log X (b) Let Y = log X Use the mgf technique (with t <r) to show that: Y Exp(1 = r) [ HINT: My(t) = Eletbox] = E[X“) = * **f(x)dt...
6. Suppose that the statistical model is given by fo(1) fo(2) fo(3) fol(4) θ=a11/3 1/6 1/3 1/6 0-b 1/21/4 1/8 1/8 and that the priors is given by π(a) = 1/3,π(b) = 2/3 and we observe the sample (x1両,xs) = (1, i, 3). (a) (5 marks) Determine the posterior of (c) (5 marks) Determine the MAP estimate of θ (d) (5 marks) Determine the relative belief estimate of θ
6. Suppose that the statistical model is given by fo(1) fo(2)...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
4.(120) Let X1,,,Xn be iid r(, 1) and g(u) given. Let 6n be the MLE of g(4) (1)(60) Find the asymptotic distribution of 6, (2)(60) Find the ARE of T Icc(X) w.r.t. on P(X1> c), c > 0 is i n i1 5.(80) Let X1, ,,Xn be iid with E(X1) = u and Var(X1) limiting distribution of nlog (1 +). o2. Find the where T n(X - 4)/s. - 1 -
4.(120) Let X1,,,Xn be iid r(, 1) and g(u)...
have the prior pdf 2-2) Let a (e)-0<a<e where B>1 1 Find the joint pdf fo, e) of Y and e Use f(y.0) py0)2(0) [Hint]
have the prior pdf 2-2) Let a (e)-0
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1 +0)-r(0-1) (1-2), 0 < x < 1, θ > 0. the estimator T-4 is a method of moments estimator for θ. It can be shown that the asymptotic distribution of T is Normal with ETT θ and Var(T) 0042)2 Apply the integral transform method (provide an equation that should be solved to obtain random observations from the distribution) to generate a sam ple of...
Let X1,.. ,X be a random sample from an N(p,02) distribution, where both and o are unknown. You will use the following facts for this ques- tion: Fact 1: The N(u,) pdf is J(rp. σ)- exp Fact 2 If X,x, is a random sample from a distribution with pdf of the form I-8, f( 0,0) = for specified fo, then we call and 82 > 0 location-scale parameters and (6,-0)/ is a pivotal quantity for 8, where 6, and ô,...
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