Two In-Depth Neyman Pearson Lemma Examples

Описание: Suppose X1,...,X10 are each N(0,σ^2) and we are testing the population variance H0:σ^2=1 versus H1:σ^2=2. What is the rejection region (critical region) for the best (most powerful) test when the significance level is α=0.05? We need the fact that X^2 has a gamma distribution and that the sum of X1^2,...,X10^2 also has a gamma distribution. Next, suppose the PDF of each of X1,...,Xn is f(θ)=θx^(θ-1) for x in (0,1) and is 0 otherwise. If we test H0:θ=1 versus H1:θ=2, the rejection region of the best test will take the form x1*x2*...*xn is greater than or equal to some constant. Outline how to find the constant when α=0.05 and n=2 using an integral of the joint density function (joint PDF) of X1*X2 (do an appropriate double integral and then solve an algebraic equation using technology). https://amzn.to/3rjDOoA (Probability and Statistics with Applications: A Problem Solving Text, by Asimow and Maxwell)

Applied Statistics, Class 16, Spring 2022

#HypothesisTesting #NeymanPearson #LikelihoodRatio

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(0:00) Neyman Pearson Lemma
(5:58) Example 1: Best critical region for testing a variance (alternative value greater than the null value) when the population is N(0,σ^2)
(9:36) Likelihood ratio
(15:20) Critical region (initial form)
(20:29) Critical region (final form)
(25:35) Start process to find K
(27:19) Distribution of square of a normal random variable with variance σ^2 (CDF Method)
(33:10) X^2 has a gamma distribution with α=1/2 and β=2σ^2
(36:50) Distribution of sum of Xi^2 (with moment generating function)
(39:20) Finish solving for K (use Mathematica)
(42:05) The answer (with Solve and FindRoot in Mathematica)
(44:28) Homework problem: solve for k
(47:01) Homework problem: find the power of the test (or probability of a Type 2 error)
(49:16) Example 2: Test θ for f(θ)=θx^(θ-1) (find best critical region)
(50:38) Likelihood ratio
(53:48) Critical region (initial form)
(57:07) Critical region (final form)
(58:09) Use θ0=1 and θ1=2
(59:17) HW problem: Find K when n=1 (and α=0.05)
(1:00:36) Finding K when n=2 (and α=0.05)
(1:02:14) Joint density of (X1,X2)
(1:04:13) Region of integration
(1:06:35) Limits of integration
(1:07:50) Set θ=θ0=1
(1:08:45) HW problem: solve for K with technology like Mathematica (symbolic answer involves ProductLog on Mathematica...you can use NSolve)

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