s330notes08

# S330notes08 - STATISTICS 330 Mathematical Statistics Supplementary Lecture Notes Cyntha A Struthers Dept of Statistics and Actuarial Science

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STATISTICS 330 Mathematical Statistics Supplementary Lecture Notes Cyntha A. Struthers Dept. of Statistics and Actuarial Science University of Waterloo Waterloo, Ontario, Canada Fall 2008

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Contents 1P R E V I EW 1 1 .1 Examp l e............................. 1 1 .2 Examp l 3 2R A N D OMV A R I A B L E S 5 2 .1 In t rodu c t ion. .......................... 5 2 .2 D i s c r e t eRandomV a r iab l e s .................. 6 2 .3 Con t inuou sRandomV a r l e s................. 8 2.4 Functions of a Random Variable X .............. 10 2 .5 Exp e c ta t 11 2 .6 M om en tG e ra t ingF un c t ion s ................ 14 2 .7 Lo ca t ionandS c a l eP a ram e t e r 17 3 Joint Distributions 19 3 t c t 19 3 .2 Jo in tD i s c r e t a r l e s ............... 20 3 .3 Jo tCon t a r l e s ............. 21 3 .4 Ind ep end tRandomV a r l e s................ 25 3 .5 Cond i t iona lD i s t r ibu t s ................... 27 3 .6 Jo tExp e c t s ....................... 30 3 .7 Cond i t lExp e c t ................... 32 3 .8 Jo tM e t c t 34 4 Functions of Random Variables 37 4 t c t 37 4 .2 On e - t o -On eB iv a r ia t eT ran s fo rm a t s............ 38 4 .3 M e t c t ionM e thod . ........... 41 5 Limiting or Asymptotic Distributions 45 5 .1 Conv e rg c einD i s t r t ................. 45 5.2 Convergence in Probability . 47 1
0 CONTENTS 5 .3 L im i tTh eo r em s......................... 50 6 Estimation 53 6 .1 In t rodu c t ion. .......................... 53 6.2 Method of Maximum Likelihood -On eP a ram e t e rCa s e ..................... 54 6.3 Method of Maximum Likelihood -Mu l t ipa e t e s e..................... 60 6.4 Asymptotic Properties of M.L. E s t ima t o r s-On a e t e s e............... 68 6 .5 In t e rv a lE s t a to r s....................... 69 6.6 Asymptotic Properties of M.L. E s t t o r s-Mu l t e t e s e .............. 75 6.7 Con f d en c eR eg ion 77 7H y p o t h e s i sT e s t s 8 3 7 t c t 83 7.2 Likelihood Ratio Tests for Simple Hypo th e s e s ........................... 84 7.3 Likelihood Ratio Tests for Composite e s e 87

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Chapter 1 PREVIEW The follow examples will illustrate the ideas and concepts we will study in STAT 330. 1.1 Example Thefo l low ingtab leg ive sthenumbe ro ffumb le sinagamemadeby110 Division A football teams during one weekend: No. of Fumbles: x 0 1 2 3 4 5 6 7 8 Total Obs. Frequency: f x 8 24 27 20 17 10 3 1 0 110 It is believed that a Poisson model will f tthesedatawe l l .Whym ight this be a reasonable assumption? (PROBABILITITY MODELS) If we let the random variable X = number of fumbles in a game and assume that the Poisson model is reasonable then the probability function (p.f.) of X is given by P ( X = x )= μ x e μ x ! x =0 , 1 ,... where μ is a parameter of the model which represents the mean number of fumbles in a game. (RANDOM VARIABLES, PROBABILITY FUNC- TIONS, EXPECTATION, MODEL PARAMETERS) Since μ is unknown we might estimate it using the sample mean ¯ x = 8(0) + 24(1) + ··· +1(7) 110 = 281 110 2 . 55 . (POINT ESTIMATION) The estimate ˆ μ x is the maximum likelihood (M.L.) estimate of μ . It is the value of μ which maximizes the likelihood 1
2 CHAPTER 1. PREVIEW function. (MAXIMUM LIKELIHOOD ESTIMATION) The likelihood function is the probability of the observed data as a function of the un- known parameter(s) in the model. The M.L. estimate is thus the value of μ which maximizes the probability of the observed data.

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## This note was uploaded on 04/27/2010 for the course STAT 330 taught by Professor Paulasmith during the Fall '08 term at Waterloo.

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S330notes08 - STATISTICS 330 Mathematical Statistics Supplementary Lecture Notes Cyntha A Struthers Dept of Statistics and Actuarial Science

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