STA 4502 Chapter 5

# STA 4502 Chapter 5 - 5.4 KOLMOGOROV SMIRNOV(KS TEST A...

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5.4 KOLMOGOROV - SMIRNOV (KS) TEST A DISTRIBUTION-FREE TEST FOR GENERAL DIFFERENCES In Chapter 4, we made inferences on the difference in location parameters of two populations , using two independent samples and the Wilcoxon Rank Sum Test. In this Chapter we are interested in testing the hypothesis of any difference in distributions of two populations, using two independent sample of sizes n and m, with m ≤ n ( 1 ). The hypotheses of interest are: Ho: The two populations have identical distribution vs . Ha : The two populations have different distributions. The null hypothesis is rejected when there is any difference (in location and/or dispersion and/or shape and or anything else) between the two populations. These hypotheses are tested using the Kolmogorov-Smirnoff (KS) test. Data: A total of N = n + m observations obtained by independent random samples from two populations, a population of X’s with dfF X ( x ) = P(X ≤ x ) and a population of Y’s with df F Y ( y ) = P(Y ≤ y ). The sample data are denoted by X 1 , X 2 , X 3 , …, X m and Y 1 , Y 2 , Y 3 , . .., Y n . Assumptions: A1. X 1 , X 2 , X 3 , . .., X m is a random sample (i.e., X j are independent of each other) from a continuous population, say Population 1, that has df ( 2 ), F X (t), where F X (t) = P(X ≤ t), for all t, - ∞ < t <∞, [i.e., X, are i.i.d. with df F X (t).] and Y 1 , Y 2 , Y 3 , . .., Y n is a random sample (i.e., Y j are independent of each other) from a continuous population, say Population 2, that has df G Y (t), where G Y (t) = P(Y ≤ t), for all t, - ∞ < t <∞, [i.e., Y j are i.i.d. with df G Y (t).] A2: X 1 , X 2 , X 3 , . .., X m and Y 1 , Y 2 , Y 3 , . .., Y n are independent samples. Hypotheses:

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STA 4502 Chapter 5 - 5.4 KOLMOGOROV SMIRNOV(KS TEST A...

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