Inference Theory MATH 7654
University of Memphis
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This 17 page Class Notes was uploaded by Trace Roob on Friday October 23, 2015. The Class Notes belongs to MATH 7654 at University of Memphis taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/228437/math-7654-university-of-memphis in Mathematics (M) at University of Memphis.
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Date Created: 10/23/15
Hypothesis Testing Hypothesis Testing Basics Tests for NormalBinomial distributions Most Powerful Tests MP UMP Uniformly Most Powerful Tests MLR Monotone Likelihood Ratio Likelihood Ratio Tests Sample sizepowerChi square Tests Hypothesis Testing Basics Formulation of HO vs H1 Ch 81 page 373 374 Type I vs type II error calculation Ch 83 page 382 Power function calculation Def 831 page 383 385 size a test vs level a test Def 835 and Def 836 page 385 Hypothesis Testing Basics Simple vs composite hypothesis One sidedtwo sided test P values Def 8326 page 397 Test statistics Rejection regioncritical region General test procedure g Types of Hypothesis Testing X1 X2 Xn iid pdf fxe HOGG0 vsH1GG1 simple vs simple HOGG0 vsH1eihGO simple vs composite two sided H0GSGO vsH1egtGo composite vs composite one sided g Test about Normal Distribution One sample Mean u c2 knownunknown Variance oz Two samples Difference ul uz 021 022 021 022 Ratio or equality of variances 021 and 022 4 One sample tests Mean p 02 known Z test 02 unknown t test Variance 02 x2 test Paired t test vs two sample t test which one to use Two samples tests Equality of pland p2 021 022 use pooled variance 021 022 use Welch approximation Equality of variances 021 and 022 F test can be used Important to known the relation between upperlower percentiles of F table Test about Binomial g Distributions One sample X N 39HoppoVS H13p po Test statisticcritical region Two samples X1 Bn1p1 independent of X2 Bn2p2 39 Ho p1 p2 VS H1 D1 5 p2 g Methods of Finding Tests Most Powerful Tests MP Neyman Pearson Lemma Thm 8312 page 388 Uniformly Most Powerful Tests UMP Likelihood Ratio Tests LRT Ch 82 pages 374 379 Bayesian Tests Ch 822 pages 379 380 Most Powerful Tests MP Neyman Pearson Lemma X1 X2 Xn iid pdf fxe H0GGO vsH1GG1 simple vs simple fxe Fl fxie Lex MP Test reject HO e 60 if fx eofx 603 k g Uniformly Most Powerful Tests X1 X2 Xn iid pdf fxe HOGSGO vsH1egt6o If the rejection region of the MP Test Hoeeo vsH139691gt90 is the same for any 61 then MP is UMP UMP Test does not exist for two sided Hoeeo vsH1ei396o In that case we nd unbiased test UMPU Q U M PU Tests eg 8319 page 392 Non existence of UMP test Unbiased Test eg 8310 page 387 361 2 360 for all 61 s 61 and 60 s 60 See also Fig 833 page 394 g Likelihood Ratio Tests HoezGO vs H1 6860C e so u see I Ax SUpegeo Lxe SUpe so Lxe LRT Def 821 page 375 Reject HO if xx S k Techniques needed Find MLEs under the condition 9 s 60 and e s G Get a speci c critical region g Likelihood Ratio Tests X1 X2 Xn iid pdf fxe H0GSGO vsH1egtGo supe S 90 Lxe is a constrained maximization problem Solution may be at its boundary GO supe Lx6 is a regular MLE problem MLR Monotone Likelihood Ratio Def 8316 page 391 Likelihood Ratio Ax Lx GoLx 91 For a fixed 90 and 61 if Ax is a monotone function of usually SS TTx then fx 9 has MLR property For a pdf with MLR then UMP exists H0GSGO vsH1egtGO g MLR and Exponential family Exponential family fX 9aX b9 eXpC9 dX If X1 X2 Xn iid fx 9 TX ZdXi is SS for e If Ce is monotone then fx 9 has MLR and critical region of UMP test for Ho e S 90 vs H1 9 gt 60 is of the form TX gt k g Examples of LRT eg 822 page 375 376 LRT for a normal population with known variance and unknown mean eg 823 page 376 LRT for a shifted exponential population eg 826 page 378 379 LRT for a normal population with unknown variance and unknown mean
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