Textbook: Probability and Statistical Inference by NITIS MUKHOPADHYAY
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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 th...
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 (>...
---------------------------------------------------------------------------------------------------------------------------------------------------------- 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...
8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ), derive the randomized UMP level α test for l, P-Po p reassigned oE versus H p Po where p, is a number between 0 and 1 8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ),...
ineed the answer for question 2 1. In each part, X,xX, are i.i.d. r.v.s from a distribution with unknown parameter 0. Use the Neyman-Pearson lemma to find the form of the critical region for the best test of Ho : θ = θο against Hi:0=0, where t o and θί are specified constants. Express the critical regions in their simplest forms, paying particular attention to the two cases , > e0 and θι < θο. (a) /(x; θ)-ge-for z 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 X1, X2, . . . , Xn is an iid sample of N (0, σ2 ) observations, where σ 2 > 0 is unknown. Consider testing H0 : σ 2 = σ 2 0 versus H1 : σ 2 6= σ 2 0 ; where σ 2 0 is known. (a) Derive a size α likelihood ratio test of H0 versus H1. Your rejection region should be written in terms of a sufficient statistic. (b) When the null...
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X is...
Concept Check: Terminology 0/3 points (graded) Suppose you observe iid samples X1,…,Xn∼P from some unknown distribution P. Let F denote a parametric family of probability distributions (for example, F could be the family of normal distributions {N(μ,σ2)}μ∈R,σ2>0). In the topic of goodness of fit testing, our goal is to answer the question "Does P belong to the family F, or is P any distribution outside of F ?" Parametric hypothesis testing is a particular case of goodness of fit testing...
(b) . Write the k-th step of the trapezoidal method as a root-finding problem Ğ = is Y+1 where the unknown (e)Find the Jacobian matrix of the vector function from the previous part. (dWrite a function in its own file with definition [Y] dampedPendulum(L, T) function alpha, beta, d, h, that approximates the solution to the equivalent system you derived in part (a) with L: the length of the pendulum string alpha: the initial displacement beta: the initial velocity d:...