44 Let X,..., X. be a random sample from Find the Pitman estimator for the location parameter (f) Using the prior densi...
Letter f and g only. 44 Let X,..., X. be a random sample from (a) Find a sufficient statistic. (b) Find a maximum-likelihood estimator of θ. (c) Find a method-of-moments estimator of θ. (d) Is there a complete sufficient statistic? If so, find it. (e) Find the UMVUE of 0 if one exists. (f) Find the Pitman estimator for the location parameter θ. (g) Using the prior density g(0)--e-n,๑)(8), find the posterior Bayes estimator Of θ. 44 Let X,..., X....
Let the random sample X1, . . . , Xn be taken from the Binomial distribution with parameter θ, which is unknown and must be estimated. Let the prior distribution of θ be the beta distribution with known parameters α > 0 and β > 0. Find the Bayes risk and the Bayes estimator using squared error loss. estimator of θ.
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the pitman estimator of 0 b. Show that the pitman estimator has smaller risk than the UMVUE of when the loss function is (t-0)2 02 Suppose C. f(x)= 0exp(-0x) and that 0 has a gamma prior with parameters a and p, find the Bayes estimator of 0 d. Find the minimum Bayes risk e. Find the minimax estimator of 0 if one exists. 1 Let...
Let Xi , i = 1, · · · , n be a random sample from Poisson(θ) with pdf f(x|θ) = e −θ θ x x! , x = 0, 1, 2, · · · . (a) Find the posterior distribution for θ when the prior is an exponential distribution with mean 1; (b) Find the Bayesian estimator under the square loss function. (c) Find a 95% credible interval for the parameter θ for the sample x1 = 2, x2...
Please answer the following question and show every step. Thank you. Let Xi,..,Xn be a random sample from a population with pdf 0, x<0, where θ > 0 is unknown. (a) Show that the Gamma(a, b) prior with pdf 0, θ < 0. is a conjugate prior for θ (a > 0 and b > 0 are known constants). (b) Find the Bayes estimator of θ under square error loss. (c) Find the Bayes estimator of (2π-10)1/2 under square error...
Let X1, . . . , Xn be independent Poisson(θ) random variables with parameter θ > 0. (1) Find the Bayes estimator of θ for a Gamma(α, β) prior. (2) Find the MSE of the Bayes estimator.
Let X ...., X., be a random sample from a geometric distribution with parameter , and let the prior distribution of be uniform on the interval from zero to one. 1. (70 Points) Find the density of the posterior distribution, including the constant that makes it integrate to one. Name the distribution.
Let X1, . . . , Xn be independent Beta(θ, 1) random variables with parameter θ > 0. (1) Find the Bayes estimator of θ for a Gamma(α, β) prior. (2) Find the MSE of the Bayes estimator.
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
2. 20 marks] Let z1,., xn be a random sample drawn independently from a one-parameter curved normal distribution which has density -oo < x < 00, θ>0, , riid i.e., X r, and 2,2-1 Γη (e) 3 marks Find the Fisher information Z(0) (f) [3 marks] Is θ2 an MVUE of θ? Justify your answer (g) 3 marks] Assume that T = 1.32 and x-3.76 for a random sample of size n = 100. Find the Wald 95% confidence interval...