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# Class Note for EMSE 273 with Professor Dorp at GW (7)

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This 5 page Class Notes was uploaded by an elite notetaker on Saturday February 7, 2015. The Class Notes belongs to a course at George Washington University taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/07/15
Extra Notes on ChiSquare Test ChiSquare GoodnessofFit Test The Chi Square test compares the empirical histogram density constructed from sample data to a candidate theoretical density Assume that the empirical sample 11 inn is a set of n iid realizations from an underlying unknown random variable X This sample is then used to construct an empirical histogram with m bins where Bin j corresponds to the interval LB j U 33 The chi square test allows some exibility in the choice on bin boundaries and the number of bins m The estimator of the probability pj P rX 6 L333 of cell is A amp 39 1 m pjN77 7 7 where N j is the number of observations in Bin These can be determined using the FREQUENCY array function in Micro Soft Excel EMSE 273 Notes by Dr JR van Dorp Page 1 Extra Notes on ChiSquare Test Let F be some theoretical candidate distribution with parameter vector Q of the random variable X Whose goodness of fit is to be assessed Then Pj PTX 6 L333 Ule FXUBjQ FXLBjQa J39 1a am Define next Oj Number of Observations in Bin j X N Ej Expected Number of Observations in Bin pj X N and m 0E2 S2 3 3 gt0 Intuition If F X is a good fit then the theoretical value of pj should be close to the estimated value 3j and thus Oj should be close to Ej Hence a good fit would have a small S2value EMSE 273 Notes by Dr JR van Dorp Page 2 Extra Notes on ChiSquare Test It can be shown that S2 is a realization of xivariable ie a chi squared random variable with k degrees of freedom where km Q 1 Here is I Q equal to the number of parameters in the vector Q Note that xi is a random variable with support 0 00 ie it only takes on non negative values Using the CHIDIST function in Microsoft Excel we can calculate the probability that xi is greater than the observed value S2 If this probability is small large than clearly the observed value 32 may be considered quotbigquot quotsmallquot Define the p value of the Chi Squared goodness of fit test as PTX gt S2 E p value EMSE 273 Notes by Dr JR van Dorp Page 3 Extra Notes on ChiSquare Test Chi squared with parameter 10 2 1 7 degrees of freedom Signi cance 1000 014 1 1 012 77777777 17 77777777 77777777777 77777777777 77777777777 01 W LL 008 77777777 7 2r 777777777 quot4 7777777777 quot4 77777777777 quot1 77777777777 H e rrrrrrrr a 0000000000 0000000000 00000000000 00000000000 h 004 H 002 777777777 77777 777777777 77777777777 77777777777 0 l l quot l o 44 1o 1202X 5 20 25 CHIquot2PDF 01 Criticality Threshold o CHIquot2 Value p valuePrS2 gt44733 PrS2 gt120210 The significance level 04 10 defines the criticality threshold 0010 Fail to Reject Theoretical Distribution ltgt S2 lt Cp ltgt p value gt 04 EMSE 273 Notes by Dr JR van Dorp Page 4 Extra Notes on ChiSquare Test It is common to reject the candidate theoretical distribution when the p value is smaller than 001 005 or even 010 Rule of thumb for the size of E j which allows the Chi Squared distribution assumption There is no real agreement on this issue it has been suggested that Ej gt 3 4 or 5 Rule of thumb for the number of Bins Sample Size N Number of Bins 20 Do not use X2 Test 50 5 to 10 100 10 to 20 gt 100 N to The Chi Square Test allows for some exibility in the choice of bin boundaries Some have suggested that boundaries should be selected such that the expected number of observations is the same in each bin This would weigh each part of the theoretical distribution equally in the chi squared fit This can be achieved by using the inverse cumulative distribution function of the theoretical distribution if available in a closed form EMSE 273 Notes by Dr JR van Dorp Page 5

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