8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where ...
Textbook: Probability and Statistical Inference by NITIS MUKHOPADHYAY 8.4.8 Suppose that.. are iid having the Uniform(0, 6) distribution with unknown 6(> 0). With preassigned α e (0, 1), derive the UMP level α test for H° : θ > versus H1 : θ 60 is a positive number in the simplest implementable form. (i) Use the Neyman-Pearson approach (ii) Use the MLR approach 8.4.8 Suppose that.. are iid having the Uniform(0, 6) distribution with unknown 6(> 0). With preassigned α...
---------------------------------------------------------------------------------------------------------------------------------------------------------- Reference: 11.2.3 Suppose that X. , X are iid MI, σ2) where σ (E 9t*) is the unknown parameter but μ(€ 9) is assumed known. With preassigned α ε (0. 1), derive a level α LR test for a null hypothesis Ho : σ.-a> 0) against an alternative hypothesis H, : σ2 σ1 in the implementable form. {Note: Recall from the Exercise 8.5.5 that no UMP level a test exists for testing Ho versus 8.5.5 Let X, X, be...
N.B: Solve only for (i), but draw the power function for (i) as well. X, are iid random variables from the M11, σ2) unknown, μ 01 8.3.1 Suppose that X1, population where μ is assumed known but σ is fix a number α E (0, 1) and to positive numbers σο, σ1. g, () ( : M". Wi Derive the MP level a test for Ho : σ in the simplest implementable form: (i) σ0 versus H1 : σ-01 (>...
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where θ > 0. (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
2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; = 1) = 01, P(Y; = 2) = 62, where the parameter vector 6 = (60,61,62) satisfies: 0; > 0 and 200; = 1. (a) Calculate E[Y] and EY?), and use the results to derive a method of moments estimator for the parameters (01,02). (b) Show that the maximum likelihood estimator for 6 = 60, 61, 62) is - Ôno = ôz = = 1(Y;=0),...
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...
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...
If X ~ N(0, σ2), then Y function of Y is X follows a half-normal distribution; i.e., the probability density This population level model might arise, for example, if X measures some type of zero-mean difference (e.g., predicted outcome from actual outcome) and we are interested in absolute differences. Suppose that Yi, ½, ,y, is an iid sample from fy(ylơ2) (a) Derive the uniformly most powerful (UMP) level α test of 2 2 0 versus Identify all critical values associated...
DE Suppose the IID random sample is (X,Y) where X, and Y, are independent random variables having Normal(11.1) and Normal(j2, 1) densities. So for X f( 1) = .7 exp f(y H1) = 7exp (yi - 12) For the following two free response questions, identify the density and the parameters of distributions of each quantity. X/01 – 2 n(x - H)2 + (n - 1)
Suppose that X1, X2,..., Xn are iid from where a 1 is a known constant and θ > 0 is an unknown parameter. (a) Show that the likelihood ratio rejection region for testing Ho : θ θο versus H : θ > θο can be written in terms of X(n), the maximum order statistic. (b) Derive the power function of the test in part (a). (c) Derive the most powerful (MP) level α test of Ho : θ-5 versus H1...