11_18_11 - STAT 409 Fall 2011 Examples for 11/18/2011 X1,...

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Examples for 11/18/2011 Fall 2011 X 1 , X 2 , … , X n Empirical Distribution Function: S ( x ) = [ number of X i ’s that are x ] / n H 0 : X 1 , X 2 , … , X n are i.i.d. with c.d.f. F ( x ) Test Statistic D = ( ) ( ) F S sup x x x - . 1. Consider a data set: 34 62 70 15 77 94 50 42 132 85 100 28 55 66 120 50 112 20 8 145 Consider H 0 : X 1 , X 2 , … , X 17 are i.i.d. Gamma with α = 5 and θ = 10. S ( x ) = [ number of X i ’s that are x ] / n = [ number of X i ’s that are x ] / 20 )/20 = 1, 2, … , 20. S ( x- ) = [ number of X i ’s that are < x ] / n = [ number of X i ’s that are < x ] / 20 )/20 = 1, 2, … , 20. If T has a Gamma ( α , θ = 1 / λ ) distribution, where α is an integer, then 2 T / θ = 2 λ T has a χ 2 ( 2 α ) distribution ( a chi-square distribution with 2 α degrees of freedom ). F
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11_18_11 - STAT 409 Fall 2011 Examples for 11/18/2011 X1,...

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