Transcript 2015_9_Hypothesis tests (3)

STATISTICS
HYPOTHESES TEST (III)
Nonparametric Goodness-of-fit (GOF) tests
Professor Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University
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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• 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 pi denote the probability that
an observation will belong to category i (i = 1,
2, …, k). Of course, pi  0 for i = 1, 2, …, k
k
and  pi  1 .
i 1
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• Therefore, it seems reasonable to base a test on
the values of the differences ni  ei
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 Dn , 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 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)
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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, alternative=”two.sided”)
ks.test(x, ”pexp”, 0.2, alternative=”greater”)
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