Let X1, ..., Xn be a random sample from Gamma(1,41) distribution and Y1, ..., Ym be...
Let X1 ,……, Xn be a random sample from a Gamma(α,β) distribution, α> 0; β> 0. Show that T = (∑n i=1 Xi, ∏ n i=1 Xi) is complete and sufficient for (α, β).
Let X1, , Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β-A-Derive the Score test for testing Ho- Let X1, , Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β-A-Derive the Score test for testing Ho-
May 21, 2019 R 3+3+5-11 points) (a) Let X1,X2, . . Xn be a random sample from G distribution. Show that T(Xi, . . . , x,)-IT-i xi is a sufficient statistic for a (Justify your work). (b) Is Uniform(0,0) a complete family? Explain why or why not (Justify your work) (c) Let X1, X2, . .., Xn denote a random sample of size n >1 from Exponential(A). Prove that (n - 1)/1X, is the MVUE of A. (Show steps.)....
Let X1, . . . , Xn ∼ iid Exp(λ) and Y1, . . . , Ym ∼ iid Exp(τ ) be independent random samples. (a) Find the restricted MLEs under the null hypothesis H0 : λ = τ . (b) Write out a formula for the LRT statistic, and describe how you could perform this test asymptotically.
Given two independent random samples X1, ..., Xn and Y1, ..., Ym with normal dis- tributions N(Hz, o?) and N(Hy, oz), determine a generalized likelihood ratio test for Ho : Mix - My = 0 versus H : plz – My 70 at a given significance level a (01, 0y unknown but equal).
1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... < Yn be the order statistics from this sample. a) Find the joint pdf of (Y1, .. , Yn). b) Find the joint pdf of (W1, .. , Wn) where W1 = nY1, W2 = (n-1)(Y2 -Y1), W3 = (n - 2)(Y3 - Y2),..., Wn-1 = 2(Yn-1 - Yn-2), Wn = Yn - Yn-1. (c) Show that Wi's are independent and its distribution is identically...
Let X1, , xn be a random sample gamma(a, β). In parts (a)-(d) assume a is known. Consider testing Ho : β Derive Wald statistic for testing Ho using the MLE of β both in the numerator and denominator of the statistic. Let X1, , xn be a random sample gamma(a, β). In parts (a)-(d) assume a is known. Consider testing Ho : β Derive Wald statistic for testing Ho using the MLE of β both in the numerator and...
Q. 4 Let X1, ... , Xn be a random sample from gamma (2,0) distribution. c) Show that it is the regular case of the exponential class of distributions. d) Find a complete, sufficient statistic for 0. e) Find the unique MVUE of 0. Justify each step.
Let X1, ,Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β = βο. Derive Wald statistic for testing Ho using the MLE of B both in the numerator and denominator of the statistic. Let X1, ,Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β = βο. Derive Wald statistic for testing Ho using the MLE of B both in the numerator and denominator of the statistic.
Let X1, ..., Xn and Y1, ..., Ym be two independent samples from a Poisson dis- tribution with parameter 1. Let a, b be two positive numbers. Consider the following estimator for 1: i ,Y1 +...+Ym = a- X1 +...+Xn n т (a) What condition is needed on a and b so that û is unbiased? (b) What is the MSE of i?