lec4 - Statistical Inference for FE Professor S. Kou,...

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Statistical Inference for FE Professor S. Kou, Department of IEOR, Columbia University Lecture 4. Goodness-of-Fit Tests In this lecture we shall study various goodness-of- f t tests, whose objec- tive is to test whether a model f ts data. We shall introduce two general chi-square tests for discrete observations, and various special tests for hy- pothesis testing on distributions. 1C h i - S q u a r e T e s t s To understand the motivation of the chi-square test we f rst consider a simple example. Example 1. The following table presents the number of trades (buy and sell stocks) conducted within a week period from 48 individual brokerage accounts. # o f t r a d e s012 3 4 F r e q u e n c y 991 01 46 For example, there are 10 accounts which had made 2 trades during the week. We want to test whether a Poisson distribution can f t the data. In general, suppose the data has been grouped into m categories (or cells); in Example 1, we have 5 cells. We want to test H 0 : p = p ( θ ) ω 0 ,H a : p 6 = p ( θ ) ,p , where θ a parameter in a given probability distribution, = { p : P m i =1 p i = 1 i 0 } . For example, in Example 1, p ( θ ) represents probabilities accord- ing to a Poisson distribution with an unknown parameter (which is λ in the Poisson distribution). 1.1 Likelihood Ratio Chi-square Test We shall use the likelihood ratio statistic, which is given by Λ = max H 0 lik ( p ( θ )) max H a lik ( p ) = max p ω 0 lik ( p ( θ )) max p lik ( p ) , where lik denotes the likelihood. Since the likelihood for the m ce llsisg iven by n ! X 1 ! ··· X m ! p X 1 1 p X 2 2 p X m m , we have Λ = max θ ω 0 n ! X 1 ! ··· X m ! p 1 ( θ ) X 1 p 2 ( θ ) X 2 p m ( θ ) X m max p n ! X 1 ! ··· X m ! p X 1 1 p X 2 2 p X m m . The denominator is maximized at ˆ p i = X i n . 1
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For the numerator, let ˆ θ be the maximizer of the numerator. Thus, the likelihood ratio test statistics is Λ = p 1 ( ˆ θ ) X 1 p 2 ( ˆ θ ) X 2 ··· p m ( ˆ θ ) X m ˆ p X 1 1 ˆ p X 2 2 ˆ p X m m = m Y i =1 à p i ( ˆ θ ) ˆ p i ! X i , and 2log Λ = 2 m X i =1 X i log à p i ( ˆ θ ) ˆ p i ! . Denote O i = X i the observed cell counts, E i = np i ( ˆ θ ) the expected cell counts under H 0 . Then we know from the general theory of likelihood ratio test that under H 0 , Λ =2 m X i =1 O i log μ O i E i approximately has χ 2 distribution with d.f. d.f. =dim( ) dim( H 0 )= m 1 dim( H 0 ) . 1.2 Pearson’s χ 2 Test To simplify the likelihood test, note the Taylor series x log( x x 0 )=( x x 0 )+ 1 2 ( x x 0 ) 2 x 0 + Therefore, we also have Λ m X i =1 O i log μ O i E i m X i =1 " ( O i E i 1 2 ( O i E i ) 2 E i + # 2 m X i =1 " ( O i E i 1 2 ( O i E i ) 2 E i # = m X i =1 ( O i E i ) 2 E i . This is called Pearson’s χ 2 test. The likelihood ratio χ 2 test is approxi- mately equivalent to Pearson’s χ 2 test. In particular, Pearson’s χ 2 statistic is approximately χ 2 with df = m 1 dim( H 0 ) Remarks: (i) In deriving the likelihood ratio χ 2 test and Pearson’s χ 2 test we assume that the unknown parameter θ is estimated by the MLE.
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This note was uploaded on 10/18/2010 for the course IEOR 4702 taught by Professor Kou during the Spring '10 term at Columbia.

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lec4 - Statistical Inference for FE Professor S. Kou,...

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