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IEOR 165 Lecture 11 June 17 2009 COPY 1

IEOR 165 Lecture 11 June 17 2009 COPY 1 - Goodness of fi t...

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Goodness of fit test CASE 1: Parameters are KNOWN Random variable Y Sample data ,…, Y1 Yn - First case: discrete case Y takes values from ,…, y1 yk We are going to test whether the probability associated with each y is true Null Hypothesis: : = = , = … H0 PY yi pi i 1 k : = = … H1 PY y1 pi i 1 k We want to fit Y1 Yn into the diagram = = = # = m1 i 1n1Yi y1 of sample data to y1 = = = mk i 1n1Yi yk Y1 y2 …….. yk Test statistics: = = - = ~ - T0 i 1kmi npi2npi where Emi npi when n is large T0 xk 12 , If the test statistic T0 is large then the numerator is large in relationship .→ to denominator bad estimation , - Since this is a x2estimation we can calculate p values Method 1: - = - p value Pxk 12 T0 the larger the p value the better the estimation ≥ - Reject H0at any significance level p value Method 2: Confidence intervals .( , - Reject H0if T0is large reject H0if T0 xα k 12 → - % : ( , , - 1 α100 CI 0 xα k 12
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- Second case: Continuous Case Y takes value in interval , 0 yk Null hypothesis : - , = = … H0 PY is inside yi 1 yi pi i 1 k Alternative hypothesis: : - ,
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