On the kolmogorov-smirnov test for normality with mean and variance unknown

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On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown Hubert W. Lilliefors Journal of the American Statistical Association, Vol. 62, No. 318. (Jun., 1967), pp. 399-402.
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ON T H E KOLR4OGOROV-SR4IRNOV TEST FOR NORRIALITY
WITH MEAN AND VARIANCE UNKNOWN

HUBERT LILLIEFORS W. The George Washington University
The standard tables used for the Kolmogorov-Smirnov test are valid when testing whether a set of observations are from a completely specified continuous distribution. If oneor more parameters must be estimated from the sample then the tables are no longer valid. A table is given in this note for use with the Kolmogorov-Smirnov statistic for testing whether a set of observations is from a normal population when the mean and variance are not specified but must be estimated from the sample. The table is obtained from a Monte Carlo calculation. A brief Monte Carloinvestigation is made of the power of the test.

Kolmogorov-Smirnov statistic provides a means of testing whether a set of observations are from some completely specified continuous distribution, F o ( X ) . The usual alternative would be the chi-square test. The Kolmogorov-Smirnov test has a t least two major advantages over the chisquare test [ref. 1, 21.

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1. It can be used with smallsample sizes, where the validity of the chisquare test would be questionable. 2. Often it appears to be a more powerful test than the chi-square test for any sample size. Unfortunately, when certain parameters of the distribution niust be estimated from the sample, then the Kolmogorov-Smirnov test no longer appliesat least not using the commonly tabulated critical points. I t is suggested in ref. 2that if the test is used in this case, the results will be conservative in the sense that the probability of a type I error will be smaller than as given by tables of the Kolmogorov-Smirnoo statistic [as found in ref. 2 or 41. As will be seen below, the results of this procedure will indeed be extremely conservative. In ref. 1 it is pointed out that if the parameters to be estimated areparameters of scale or location, then one can construct tables for use with the Kolmogorov-Smirnov statistic for that particular distribution. This note presents a table for use with the Kolmogorov-Smirnov statistic when testing that a set of observations are from a normal population but the mean and variance are not specified. The procedure is: Given a sample of N observations, one determines D =maxx /...
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