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253 42. Suppose that Xn B(n, p), and let p X/n. The CLT implies that se...
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0....
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo,...
Finish the proof of Theorem 3.14.
Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo, and z" →-oo. . If -oo <x o, then +yn 2. If-oo x < 00, then x" + Zn →-00 4. If-oo x < 0, then xoY" →-00 and xnZn → oo. 5. If x is in R. then-→0and-" →0 Proof Note that the conditions in the different parts of the...
Valu Exercises 8.2. x, . . . ,x, nd G(p), the geometric distribution with mean 1/p. Assume that e...
valu Exercises 8.2. x, . . . ,x, nd G(p), the geometric distribution with mean 1/p. Assume that e size n is sufficiently large to warrant to invocation of the Central Limit Theo- . Suppose se that Xi , . . . X, Use the asymptotic d confidence interval for p Suppose that XN(0, o2) (a) Obtain the asymptotic distribution of the second istribution of p 1/X to obtain an approximate 100(1-u)% Suppose sample moment m2 -(I/n)i X. (b) Identify...
Let X1,X2, variance ơ2. Suppose that n < 30 (and the CLT is not applicable). Use...
Let X1,X2, variance ơ2. Suppose that n < 30 (and the CLT is not applicable). Use Chebyshev's inequality for X to construct a 90% CI for μ. Discuss a disadvantage of this method when it is additionally known that the population is normal. 9. , Xn be a random sample from a population with an unknown μ and known
8. Let X,.. , Xn be a random sampl le from a uniform(O, 0) distributio n....
8. Let X,.. , Xn be a random sampl le from a uniform(O, 0) distributio n. (a) Write down the likelihood function of (b) Suppose the prior distribution of θ is given by the Pareto(α, β) distribution with pdf αβα θα+1 , for θ > β > 0, α > 0 Derive the posterior distribution of 0 and conclude that the Pareto family of distributions is a conjugate prior for the uniform distribution.
Let {Xn} be a sequence of RVs with Xn~G(n,β), where β>0 is a constant (independent of...
Let {Xn} be a sequence of RVs with Xn~G(n,β), where β>0 is a constant (independent of n). Find the limiting distribution of Xn/n.
Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0...
Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0 < a < 0o. (a) Find the MLE of 0. (b) Let T = -log X. Find the pdf of T. (c) Find the pdf of Y = DIT: (i.e., distribution of Y = - , log Xi). (d) Find E(). (e) Find E( ). (f) Show that the variance of 0 MLE → as n → 00. (g) Find the MME of 0.
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1),...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
Part 1: Derive the expected value and find the asymptotic distribution. Part 2: Find the consistent estimator and use the central limit theorem b. Derive the expected value of X for the Weibull(X,2)...
Part 1: Derive the expected value and find the asymptotic
distribution.
Part 2: Find the consistent estimator and use the central limit
theorem
b. Derive the expected value of X for the Weibull(X,2) distribution. c. Suppose X,.. .X,~iid Uniffo,0). Find the asymptotic distribution of Z-n(-Xm) max Let X, X, ~İ.id. Exp(β). a. Find a consistent estimator for the second moment E(X"). Use the mgf of X to prove that your estimator is consistent in the case β=2 b. Use the...
- Let X1, X2, ..., Xn be iid from the pdf fe(x) = 0e-82, > 0....
- Let X1, X2, ..., Xn be iid from the pdf fe(x) = 0e-82, > 0. Note that T = 2 , X, is a sufficient statistic. Consider testing the hypothesis H.: 8 = 1 vs H: 8 = 2 using Bayes method. Suppose the prior distribution is P(0 = 1) = ? and P(0 = 2) = 1 - . (a) Show that the Bayes test rejects H, if T < In log(2) + log((1 - ))) (b) Take...
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Finish the proof of Theorem 3.14.
Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo, and z" →-oo. . If -oo <x o, then +yn 2. If-oo x < 00, then x" + Zn →-00 4. If-oo x < 0, then xoY" →-00 and xnZn → oo. 5. If x is in R. then-→0and-" →0 Proof Note that the conditions in the different parts of the...
valu Exercises 8.2. x, . . . ,x, nd G(p), the geometric distribution with mean 1/p. Assume that e size n is sufficiently large to warrant to invocation of the Central Limit Theo- . Suppose se that Xi , . . . X, Use the asymptotic d confidence interval for p Suppose that XN(0, o2) (a) Obtain the asymptotic distribution of the second istribution of p 1/X to obtain an approximate 100(1-u)% Suppose sample moment m2 -(I/n)i X. (b) Identify...
Let X1,X2, variance ơ2. Suppose that n < 30 (and the CLT is not applicable). Use Chebyshev's inequality for X to construct a 90% CI for μ. Discuss a disadvantage of this method when it is additionally known that the population is normal. 9. , Xn be a random sample from a population with an unknown μ and known
8. Let X,.. , Xn be a random sampl le from a uniform(O, 0) distributio n. (a) Write down the likelihood function of (b) Suppose the prior distribution of θ is given by the Pareto(α, β) distribution with pdf αβα θα+1 , for θ > β > 0, α > 0 Derive the posterior distribution of 0 and conclude that the Pareto family of distributions is a conjugate prior for the uniform distribution.
Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0 < a < 0o. (a) Find the MLE of 0. (b) Let T = -log X. Find the pdf of T. (c) Find the pdf of Y = DIT: (i.e., distribution of Y = - , log Xi). (d) Find E(). (e) Find E( ). (f) Show that the variance of 0 MLE → as n → 00. (g) Find the MME of 0.
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
Part 1: Derive the expected value and find the asymptotic
distribution.
Part 2: Find the consistent estimator and use the central limit
theorem
b. Derive the expected value of X for the Weibull(X,2) distribution. c. Suppose X,.. .X,~iid Uniffo,0). Find the asymptotic distribution of Z-n(-Xm) max Let X, X, ~İ.id. Exp(β). a. Find a consistent estimator for the second moment E(X"). Use the mgf of X to prove that your estimator is consistent in the case β=2 b. Use the...
- Let X1, X2, ..., Xn be iid from the pdf fe(x) = 0e-82, > 0. Note that T = 2 , X, is a sufficient statistic. Consider testing the hypothesis H.: 8 = 1 vs H: 8 = 2 using Bayes method. Suppose the prior distribution is P(0 = 1) = ? and P(0 = 2) = 1 - . (a) Show that the Bayes test rejects H, if T < In log(2) + log((1 - ))) (b) Take...
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