Transcription of Likelihood Ratio Tests - Missouri State University
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Likelihood Ratio Tests - Missouri State University
Math 541: Statistical Theory IILikelihood Ratio TestsInstructor: Songfeng ZhengA very popular form of hypothesis test is the Likelihood Ratio test, which is a generalization ofthe optimal test for simple null and alternative hypotheses that was developed by Neymanand Pearson (We skipped Neyman-Pearson lemma because we are short of time). Thelikelihood Ratio test is based on the Likelihood functionfn(X 1, , Xn| ), and the intuitionthat the Likelihood function tends to be highest near the true value of . Indeed, this is alsothe foundation for maximum Likelihood estimation. We will start from a very simple The Simplest Case: Simple HypothesesLet us first consider the simple hypotheses in which both the null hypothesis and alternativehypothesis consist one value of the parameter. SupposeX1, , Xnis a random sample ofsizenfrom an exponential distributionf(x| ) =1 e x/ ;x >0 Conduct the following simple hypothesis testing problem:H0: = 0vs. Ha: = 1,where 1< 0. Suppose the significant level is.
2 Intuitively, if the evidence (data) supports H1, then the likelihood function fn(X1;¢¢¢;Xnjµ1) should be large, therefore the likelihood ratio is small. Thus, we reject the null hypothesis if the likelihood ratio is small, i.e. LR • k, where k is a constant such that P(LR • k) = fi under the null hypothesis (µ = µ0).To flnd what kind of test results from this criterion, we expand ...
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