STATISTICS HYPOTHESES TEST (III) Nonparametric Goodness-of-fit (GOF) tests

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STATISTICSHYPOTHESES TEST (III)Nonparametric
Goodness-of-fit (GOF) tests

• Professor Ke-Sheng Cheng
• Department of Bioenvironmental Systems
  Engineering
• National Taiwan University

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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.

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• 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.

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• 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.

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• 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.

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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.

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• Lets 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 pi denote
  the probability that an observation will belong
  to category i (i 1, 2, , k). Of course, for
  i 1, 2, , k and .

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• Therefore, it seems reasonable to base a test on
  the values of the differences
• for i 1, 2, , k and reject Ho when the
  magnitudes of these differences are relatively
  large.

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Chi-square GOF test
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Number of categories
Sample size
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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.

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• In order to measure the difference between Fn(X)
  and F(X), ECDF statistics based on the vertical
  distances between Fn(X) and F(X) have been
  proposed.

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Values of for the Kolmogorov-Smirnov test
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Goodness-of-fit tests using R

• ?2 test for GOF test
• chisq.test
• The above test doesnt 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)

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• 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, alternativetwo.side
  d)
• ks.test(x, pexp, 0.2, alternativegreater)