1. STATISTICS HYPOTHESES TEST (III) Nonparametric Goodness-of-fit (GOF) tests Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. Description of nonparametric Problems Until now, in the estimation and hypotheses testing problems, we have assumed that the available observations come from distributions for which the exact form is known, even though the values of some parameters are unknown. In other words, we have assumed that the observations come from a certain parametric family of distributions, and a statistical inference must be made about the values of the parameters defining that family. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 2

3. In many situations, we do not assume that the available observations come from a particular family of distributions. Instead, we want to study inferences that can be made about the distribution from which the observations come, without making special assumptions about the form of that distribution. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 3

4. For example, we might simply assume that observations form a random sample from a continuous distribution, without specifying the form of this distribution any further; and we then investigate the possibility that this distribution is a normal distribution. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 4

5. Problems in which the possible distributions of the observations are not restricted to a specific parametric family are called nonparametric problems, and the statistical methods that are applicable in such problems are called nonparametric methods. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 5

6. Goodness-of-fit test A very common statistical problem in hydrological frequency analysis or water resources planning is that whether the available observations (a random sample available to us) come from a particular type of distribution. For example, before we can estimate the magnitude of the 24-hour rainfall depth with 100-year return period, we must decide (identify) the type of probability distribution for the rainfall data (the annual maximum series) through statistical tests. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 6

7. Let s consider statistical problems based on data such that each observation can be classified as belonging to one of a finite number of possible categories. If a large population consists of data of k different categories, and let p i denote the probability that an observation will belong to category i (i = 1, 2, …, k). Of course, for i = 1, 2, …, k and. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 7

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11. Therefore, it seems reasonable to base a test on the values of the differences for i = 1, 2, …, k and reject H o when the magnitudes of these differences are relatively large. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 11

12. Chi-square GOF test 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 12

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18. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 18 Sample size Number of categories

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20. Kolmogorov-Smirnov GOF test The chi-square test compares the empirical histogram against the theoretical histogram. In contrast, the K-S test compares the empirical cumulative distribution function (ECDF) against the theoretical CDF. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 20

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24. In order to measure the difference between F n (X) and F(X), ECDF statistics based on the vertical distances between F n (X) and F(X) have been proposed. 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 24

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33. Goodness-of-fit tests using R 2 test for GOF test – chisq.test – The above test doesn t account for any parameters in determining the expected values. – The degree of freedom of the test statistic is k-1. Kolmogorov-Smirnov GOF test – ks.test (one-sample test) 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 33

34. ks.test(x, y, parameters, alternative= … ) where x is the data vector to be tested, y is a string vector specifying the hypothesized distribution, parameters are the values of distribution parameters corresponding to y, and alternative represents a string vector ( less, greater, or two.sided ) for one-tail or two-tail test. Examples ks.test(x, pnorm, 30, 10, alternative= two.sided ) ks.test(x, pexp, 0.2, alternative= greater ) 1/31/2014 Dept of Bioenvironmental Systems Engineering National Taiwan University 34