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 is unknown. (a) Show that the G...
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.
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise, where θ > 0. a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ (c) Find the pdf of θ in (b).
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
Problem 1.2 Let Xi, X2, ..., Xn be a random sample from the pdf a) Find the maximum likelihood estimator of. θΜΕ- b) Find the method of moments estimator of 0. NDM c) If a random sample of n - 4 yields the following data: method of moments estimate of θ would be θΜΟΜ- MOM 7.50, 3.73, 4.52, 3.35 then the maximumn likelihood estimate of θ would be éMLE-- and the
Let xi, R2, ...xn be a random from a population with pdf sample f(x) = 2x2-1 for Ocnaliaso 0 otherwise i) is the method of moments estimator for a consistent?
3. Let X1,... ,Xn be a random sample from a population with pdf 0, otherwise, where θ > 0. (a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ. (c) Find the pdf of θ in (b).
3. Let X1,... ,Xn be a random sample from a population with pdf 0, otherwise, where θ > 0. (a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ. (c) Find the pdf of θ in (b).
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
6. Let Xi,.Xn be a random sample from the pdf Find the method of moments estimator of θ.
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x < oo. (a) Find the method of moments estimator of θ. (b) Find the maxinum likelihood estimator of θ