ASSS NEYMAN - MATH4427 Notebook 3 Spring 2016 prepared by...

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MATH4427 Notebook 3 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MATH4427 Notebook 3 3 3.1 Hypotheses: Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Hypothesis Tests: Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Measuring the Performance of Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . 8 3.3.1 Parameter Space, Null Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3.2 Errors of Type I and II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.3 Size, Significance Level, and Power . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.4 Observed Significance Level, P Value . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3.5 Comparing Tests: Power for Fixed Size . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Hypothesis Tests for Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5 Hypothesis Tests for Bernoulli/Binomial Distributions . . . . . . . . . . . . . . . . . . . 19 3.6 Hypothesis Tests for Poisson Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.7 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.7.1 Likelihood Ratio Test and Neyman-Pearson Lemma For Simple Hypotheses . . . 23 3.7.2 Generalized Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.7.3 Large Sample Theory: Wilks’ Theorem . . . . . . . . . . . . . . . . . . . . . . . 30 3.7.4 Approximate Likelihood Ratio Tests; Special Cases . . . . . . . . . . . . . . . . . 30 3.7.5 Pearson’s Goodness-of-Fit Test Revisited . . . . . . . . . . . . . . . . . . . . . . 36 1
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3 MATH4427 Notebook 3 This notebook is concerned with the second core topic of mathematical statistics, namely, hypothesis testing theory. The notes include material from Chapter 9 of the Rice textbook. 3.1 Hypotheses: Definitions and Examples 1. Hypothesis: An hypothesis is an assertion about the distribution of a random variable or a random k -tuple. 2. Simple vs Compound Hypotheses: There are two types of hypotheses: A simple hypothesis specifies the distribution completely. For example, H : X is a Bernoulli random variable with parameter p = 0 . 45. H : ( X 1 , X 2 , X 3 ) has a multinomial distribution with n = 10 and ( p 1 , p 2 , p 3 ) = (0 . 3 , 0 . 5 , 0 . 2). A compound hypothesis does not specify the distribution completely. For example, H : X is a Bernoulli random variable with parameter p 0 . 45. H : ( X 1 , X 2 , X 3 ) has a multinomial distribution with n = 10. 3. Neyman-Pearson Framework: In the Neyman-Pearson framework of hypothesis testing, there are two competing assertions: (a) The null hypothesis, H O , and (b) The alternative hypothesis, H A . The null hypothesis is accepted as true unless sufficient evidence is provided to the contrary. If sufficient evidence is provided to the contrary, then the null hypothesis is rejected in favor of the alternative hypothesis. If there is insufficient evidence to reject a null hypothesis, researchers often say that they have “failed to reject the null hypothesis” rather than they “accept the null hypothesis”. Example: Efficacy of a proposed new treatment. Suppose that the standard treatment for a given medical condition is effective in 45% of patients. A new treatment promises to be effective in more than 45% of patients. In testing the efficacy of the proposed new treatment, the hypotheses could be set up as follows: H O : The new treatment is no more effective than the standard treatment. H A : The new treatment is more effective than the standard treatment.
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