2. Suppose that Xi, , Xn, n-: 25, form a random sample from a normal distribution with mean θ and...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 which are both unknown. (a) Given observations xi, ,Xn, one would like to obtain a (1-a) x 100% one-sided confidence interval for u as a form of L E (-00, u) the expression of u for any a and n. (b) Based on part (a), use the duality between confidence interval and hypothesis testing problem, find a critical region of size...
xn be i.id. from the Pareto distribution Pa(θ, c), θ Let Xi 0 (a) Derive a ÚMP test of size α for testing Ho : θ θ when c is known. versus Hi : xn be i.id. from the Pareto distribution Pa(θ, c), θ Let Xi 0 (a) Derive a ÚMP test of size α for testing Ho : θ θ when c is known. versus Hi :
Problem 13.2 Assume that Xi, X2,. Xa form a random sample from a normal distribution for which the mean μ is unknown and the variance is 1 . Suppose the following are to be tested: H:H>0 hypotheses at the level of significance α,-0.025 and Let δ. denote the UMP test of these let π(u 18) denote the power function of the test procedure δ a) The yMP test rejects Ho when X 2 c. Determine the appropriate value for c...
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where θ > 0. (a) Find the maximum likelihood estimator (MLE) of θ (b) Give the form of the likelihood ratio test for Ho : θ-Bo versus H1: θ > θο. (c) Show that there is an appropriate statistic T - T(X) that has monotone likelihood ratio. (d) Derive the uniformly most powerful (UMP) level α test for versusS You must give an explicit expression...
Let Xi, c〉0. Xn be i.i.d. from the Pareto distribution Pa(θ.e), θ 〉 0 Derive a ÙNIP test of size α for testing Ho : θ-Bo, c co versus 0, C > Co Let Xi, c〉0. Xn be i.i.d. from the Pareto distribution Pa(θ.e), θ 〉 0 Derive a ÙNIP test of size α for testing Ho : θ-Bo, c co versus 0, C > Co
Let Xi, c〉0. Xn be i.i.d. from the Pareto distribution Pa(θ.e), θ 〉 0 Derive a ÙNIP test of size α for testing Ho : θ-Bo, c co versus 0, C > Co Let Xi, c〉0. Xn be i.i.d. from the Pareto distribution Pa(θ.e), θ 〉 0 Derive a ÙNIP test of size α for testing Ho : θ-Bo, c co versus 0, C > Co
Let Xi,, Xn be a random sample of size n from the normal distribution with mean parameter 0 and variance σ2-3. (a) Justify thatX X, has a normal distribution with mean parameter 0 and variance 3 /n, this is, X~N(0,3/m) (you can do it formally using m.g.f. or use results from normal distribution to justify (b) Find the 0.975 quantile of a standard normal distribution (you can use a table, software or internet to find the quantile). (c) Find the...
3. Let Xi, ,X, be i.id. from a normal distribution N(1,0), for θ > 0, Find a UMP test of size α for testing Ho : θ < θο versus H1 : θ > θο. 3. Let Xi, ,X, be i.id. from a normal distribution N(1,0), for θ > 0, Find a UMP test of size α for testing Ho : θ θο.
Suppose that Xi, X2, ....Xn is an iid sample from where θ 0 is unknown. (a) Find the uniformly minimum variance unbiased estimator (UM VUE) of (b) Find the uniformly most powerful (UMP) test of versuS where θο is known. (c) Derive an expression for the power function of the test in part (b) Suppose that Xi, X2, ....Xn is an iid sample from where θ 0 is unknown. (a) Find the uniformly minimum variance unbiased estimator (UM VUE) of...