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<div class="subTitle">org.apache.commons.math3.stat.inference</div>
<h2 title="Class KolmogorovSmirnovTest" class="title">Class KolmogorovSmirnovTest</h2>
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<pre>public class <span class="strong">KolmogorovSmirnovTest</span>
extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre>
<div class="block">Implementation of the <a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test">
Kolmogorov-Smirnov (K-S) test</a> for equality of continuous distributions.
<p>
The K-S test uses a statistic based on the maximum deviation of the empirical distribution of
sample data points from the distribution expected under the null hypothesis. For one-sample tests
evaluating the null hypothesis that a set of sample data points follow a given distribution, the
test statistic is \(D_n=\sup_x |F_n(x)-F(x)|\), where \(F\) is the expected distribution and
\(F_n\) is the empirical distribution of the \(n\) sample data points. The distribution of
\(D_n\) is estimated using a method based on [1] with certain quick decisions for extreme values
given in [2].
</p>
<p>
Two-sample tests are also supported, evaluating the null hypothesis that the two samples
<code>x</code> and <code>y</code> come from the same underlying distribution. In this case, the test
statistic is \(D_{n,m}=\sup_t | F_n(t)-F_m(t)|\) where \(n\) is the length of <code>x</code>, \(m\) is
the length of <code>y</code>, \(F_n\) is the empirical distribution that puts mass \(1/n\) at each of
the values in <code>x</code> and \(F_m\) is the empirical distribution of the <code>y</code> values. The
default 2-sample test method, <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovTest(double[],%20double[])"><code>kolmogorovSmirnovTest(double[], double[])</code></a> works as
follows:
<ul>
<li>For small samples (where the product of the sample sizes is less than
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#LARGE_SAMPLE_PRODUCT">10000</a>), the method presented in [4] is used to compute the
exact p-value for the 2-sample test.</li>
<li>When the product of the sample sizes exceeds <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#LARGE_SAMPLE_PRODUCT">10000</a>, the asymptotic
distribution of \(D_{n,m}\) is used. See <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#approximateP(double,%20int,%20int)"><code>approximateP(double, int, int)</code></a> for details on
the approximation.</li>
</ul></p><p>
If the product of the sample sizes is less than <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#LARGE_SAMPLE_PRODUCT">10000</a> and the sample
data contains ties, random jitter is added to the sample data to break ties before applying
the algorithm above. Alternatively, the <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#bootstrap(double[],%20double[],%20int,%20boolean)"><code>bootstrap(double[], double[], int, boolean)</code></a>
method, modeled after <a href="http://sekhon.berkeley.edu/matching/ks.boot.html">ks.boot</a>
in the R Matching package [3], can be used if ties are known to be present in the data.
</p>
<p>
In the two-sample case, \(D_{n,m}\) has a discrete distribution. This makes the p-value
associated with the null hypothesis \(H_0 : D_{n,m} \ge d \) differ from \(H_0 : D_{n,m} > d \)
by the mass of the observed value \(d\). To distinguish these, the two-sample tests use a boolean
<code>strict</code> parameter. This parameter is ignored for large samples.
</p>
<p>
The methods used by the 2-sample default implementation are also exposed directly:
<ul>
<li><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#exactP(double,%20int,%20int,%20boolean)"><code>exactP(double, int, int, boolean)</code></a> computes exact 2-sample p-values</li>
<li><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#approximateP(double,%20int,%20int)"><code>approximateP(double, int, int)</code></a> uses the asymptotic distribution The <code>boolean</code>
arguments in the first two methods allow the probability used to estimate the p-value to be
expressed using strict or non-strict inequality. See
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovTest(double[],%20double[],%20boolean)"><code>kolmogorovSmirnovTest(double[], double[], boolean)</code></a>.</li>
</ul>
</p>
<p>
References:
<ul>
<li>[1] <a href="http://www.jstatsoft.org/v08/i18/"> Evaluating Kolmogorov's Distribution</a> by
George Marsaglia, Wai Wan Tsang, and Jingbo Wang</li>
<li>[2] <a href="http://www.jstatsoft.org/v39/i11/"> Computing the Two-Sided Kolmogorov-Smirnov
Distribution</a> by Richard Simard and Pierre L'Ecuyer</li>
<li>[3] Jasjeet S. Sekhon. 2011. <a href="http://www.jstatsoft.org/article/view/v042i07">
Multivariate and Propensity Score Matching Software with Automated Balance Optimization:
The Matching package for R</a> Journal of Statistical Software, 42(7): 1-52.</li>
<li>[4] Wilcox, Rand. 2012. Introduction to Robust Estimation and Hypothesis Testing,
Chapter 5, 3rd Ed. Academic Press.</li>
</ul>
<br/>
Note that [1] contains an error in computing h, refer to <a
href="https://issues.apache.org/jira/browse/MATH-437">MATH-437</a> for details.
</p></div>
<dl><dt><span class="strong">Since:</span></dt>
<dd>3.3</dd></dl>
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<td class="colFirst"><code>protected static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#KS_SUM_CAUCHY_CRITERION">KS_SUM_CAUCHY_CRITERION</a></strong></code>
<div class="block">Convergence criterion for <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#ksSum(double,%20double,%20int)"><code>ksSum(double, double, int)</code></a></div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>protected static int</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#LARGE_SAMPLE_PRODUCT">LARGE_SAMPLE_PRODUCT</a></strong></code>
<div class="block">When product of sample sizes exceeds this value, 2-sample K-S test uses asymptotic
distribution to compute the p-value.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>protected static int</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#MAXIMUM_PARTIAL_SUM_COUNT">MAXIMUM_PARTIAL_SUM_COUNT</a></strong></code>
<div class="block">Bound on the number of partial sums in <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#ksSum(double,%20double,%20int)"><code>ksSum(double, double, int)</code></a></div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>protected static int</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#MONTE_CARLO_ITERATIONS">MONTE_CARLO_ITERATIONS</a></strong></code>
<div class="block"><strong>Deprecated.</strong> </div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>protected static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#PG_SUM_RELATIVE_ERROR">PG_SUM_RELATIVE_ERROR</a></strong></code>
<div class="block">Convergence criterion for the sums in #pelzGood(double, double, int)}</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>protected static int</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#SMALL_SAMPLE_PRODUCT">SMALL_SAMPLE_PRODUCT</a></strong></code>
<div class="block"><strong>Deprecated.</strong> </div>
</td>
</tr>
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<td class="colOne"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#KolmogorovSmirnovTest()">KolmogorovSmirnovTest</a></strong>()</code>
<div class="block">Construct a KolmogorovSmirnovTest instance with a default random data generator.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colOne"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#KolmogorovSmirnovTest(org.apache.commons.math3.random.RandomGenerator)">KolmogorovSmirnovTest</a></strong>(<a href="../../../../../../org/apache/commons/math3/random/RandomGenerator.html" title="interface in org.apache.commons.math3.random">RandomGenerator</a> rng)</code>
<div class="block"><strong>Deprecated.</strong> </div>
</td>
</tr>
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<th class="colLast" scope="col">Method and Description</th>
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<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#approximateP(double,%20int,%20int)">approximateP</a></strong>(double d,
int n,
int m)</code>
<div class="block">Uses the Kolmogorov-Smirnov distribution to approximate \(P(D_{n,m} > d)\) where \(D_{n,m}\)
is the 2-sample Kolmogorov-Smirnov statistic.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#bootstrap(double[],%20double[],%20int)">bootstrap</a></strong>(double[] x,
double[] y,
int iterations)</code>
<div class="block">Computes <code>bootstrap(x, y, iterations, true)</code>.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#bootstrap(double[],%20double[],%20int,%20boolean)">bootstrap</a></strong>(double[] x,
double[] y,
int iterations,
boolean strict)</code>
<div class="block">Estimates the <i>p-value</i> of a two-sample
<a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>x</code> and <code>y</code> are samples drawn from the same
probability distribution.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#cdf(double,%20int)">cdf</a></strong>(double d,
int n)</code>
<div class="block">Calculates \(P(D_n < d)\) using the method described in [1] with quick decisions for extreme
values given in [2] (see above).</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#cdf(double,%20int,%20boolean)">cdf</a></strong>(double d,
int n,
boolean exact)</code>
<div class="block">Calculates <code>P(D_n < d)</code> using method described in [1] with quick decisions for extreme
values given in [2] (see above).</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#cdfExact(double,%20int)">cdfExact</a></strong>(double d,
int n)</code>
<div class="block">Calculates <code>P(D_n < d)</code>.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#exactP(double,%20int,%20int,%20boolean)">exactP</a></strong>(double d,
int n,
int m,
boolean strict)</code>
<div class="block">Computes \(P(D_{n,m} > d)\) if <code>strict</code> is <code>true</code>; otherwise \(P(D_{n,m} \ge
d)\), where \(D_{n,m}\) is the 2-sample Kolmogorov-Smirnov statistic.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovStatistic(double[],%20double[])">kolmogorovSmirnovStatistic</a></strong>(double[] x,
double[] y)</code>
<div class="block">Computes the two-sample Kolmogorov-Smirnov test statistic, \(D_{n,m}=\sup_x |F_n(x)-F_m(x)|\)
where \(n\) is the length of <code>x</code>, \(m\) is the length of <code>y</code>, \(F_n\) is the
empirical distribution that puts mass \(1/n\) at each of the values in <code>x</code> and \(F_m\)
is the empirical distribution of the <code>y</code> values.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovStatistic(org.apache.commons.math3.distribution.RealDistribution,%20double[])">kolmogorovSmirnovStatistic</a></strong>(<a href="../../../../../../org/apache/commons/math3/distribution/RealDistribution.html" title="interface in org.apache.commons.math3.distribution">RealDistribution</a> distribution,
double[] data)</code>
<div class="block">Computes the one-sample Kolmogorov-Smirnov test statistic, \(D_n=\sup_x |F_n(x)-F(x)|\) where
\(F\) is the distribution (cdf) function associated with <code>distribution</code>, \(n\) is the
length of <code>data</code> and \(F_n\) is the empirical distribution that puts mass \(1/n\) at
each of the values in <code>data</code>.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovTest(double[],%20double[])">kolmogorovSmirnovTest</a></strong>(double[] x,
double[] y)</code>
<div class="block">Computes the <i>p-value</i>, or <i>observed significance level</i>, of a two-sample <a
href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>x</code> and <code>y</code> are samples drawn from the same
probability distribution.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovTest(double[],%20double[],%20boolean)">kolmogorovSmirnovTest</a></strong>(double[] x,
double[] y,
boolean strict)</code>
<div class="block">Computes the <i>p-value</i>, or <i>observed significance level</i>, of a two-sample <a
href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>x</code> and <code>y</code> are samples drawn from the same
probability distribution.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovTest(org.apache.commons.math3.distribution.RealDistribution,%20double[])">kolmogorovSmirnovTest</a></strong>(<a href="../../../../../../org/apache/commons/math3/distribution/RealDistribution.html" title="interface in org.apache.commons.math3.distribution">RealDistribution</a> distribution,
double[] data)</code>
<div class="block">Computes the <i>p-value</i>, or <i>observed significance level</i>, of a one-sample <a
href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>data</code> conforms to <code>distribution</code>.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovTest(org.apache.commons.math3.distribution.RealDistribution,%20double[],%20boolean)">kolmogorovSmirnovTest</a></strong>(<a href="../../../../../../org/apache/commons/math3/distribution/RealDistribution.html" title="interface in org.apache.commons.math3.distribution">RealDistribution</a> distribution,
double[] data,
boolean exact)</code>
<div class="block">Computes the <i>p-value</i>, or <i>observed significance level</i>, of a one-sample <a
href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>data</code> conforms to <code>distribution</code>.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovTest(org.apache.commons.math3.distribution.RealDistribution,%20double[],%20double)">kolmogorovSmirnovTest</a></strong>(<a href="../../../../../../org/apache/commons/math3/distribution/RealDistribution.html" title="interface in org.apache.commons.math3.distribution">RealDistribution</a> distribution,
double[] data,
double alpha)</code>
<div class="block">Performs a <a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov
test</a> evaluating the null hypothesis that <code>data</code> conforms to <code>distribution</code>.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#ksSum(double,%20double,%20int)">ksSum</a></strong>(double t,
double tolerance,
int maxIterations)</code>
<div class="block">Computes \( 1 + 2 \sum_{i=1}^\infty (-1)^i e^{-2 i^2 t^2} \) stopping when successive partial
sums are within <code>tolerance</code> of one another, or when <code>maxIterations</code> partial sums
have been computed.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#monteCarloP(double,%20int,%20int,%20boolean,%20int)">monteCarloP</a></strong>(double d,
int n,
int m,
boolean strict,
int iterations)</code>
<div class="block">Uses Monte Carlo simulation to approximate \(P(D_{n,m} > d)\) where \(D_{n,m}\) is the
2-sample Kolmogorov-Smirnov statistic.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#pelzGood(double,%20int)">pelzGood</a></strong>(double d,
int n)</code>
<div class="block">Computes the Pelz-Good approximation for \(P(D_n < d)\) as described in [2] in the class javadoc.</div>
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<pre>protected static final int MAXIMUM_PARTIAL_SUM_COUNT</pre>
<div class="block">Bound on the number of partial sums in <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#ksSum(double,%20double,%20int)"><code>ksSum(double, double, int)</code></a></div>
<dl><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../constant-values.html#org.apache.commons.math3.stat.inference.KolmogorovSmirnovTest.MAXIMUM_PARTIAL_SUM_COUNT">Constant Field Values</a></dd></dl>
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<pre>protected static final double KS_SUM_CAUCHY_CRITERION</pre>
<div class="block">Convergence criterion for <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#ksSum(double,%20double,%20int)"><code>ksSum(double, double, int)</code></a></div>
<dl><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../constant-values.html#org.apache.commons.math3.stat.inference.KolmogorovSmirnovTest.KS_SUM_CAUCHY_CRITERION">Constant Field Values</a></dd></dl>
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<pre>protected static final double PG_SUM_RELATIVE_ERROR</pre>
<div class="block">Convergence criterion for the sums in #pelzGood(double, double, int)}</div>
<dl><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../constant-values.html#org.apache.commons.math3.stat.inference.KolmogorovSmirnovTest.PG_SUM_RELATIVE_ERROR">Constant Field Values</a></dd></dl>
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<pre><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Deprecated.html?is-external=true" title="class or interface in java.lang">@Deprecated</a>
protected static final int SMALL_SAMPLE_PRODUCT</pre>
<div class="block"><span class="strong">Deprecated.</span> </div>
<div class="block">No longer used.</div>
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<pre>protected static final int LARGE_SAMPLE_PRODUCT</pre>
<div class="block">When product of sample sizes exceeds this value, 2-sample K-S test uses asymptotic
distribution to compute the p-value.</div>
<dl><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../constant-values.html#org.apache.commons.math3.stat.inference.KolmogorovSmirnovTest.LARGE_SAMPLE_PRODUCT">Constant Field Values</a></dd></dl>
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<pre><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Deprecated.html?is-external=true" title="class or interface in java.lang">@Deprecated</a>
protected static final int MONTE_CARLO_ITERATIONS</pre>
<div class="block"><span class="strong">Deprecated.</span> </div>
<div class="block">Default number of iterations used by <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#monteCarloP(double,%20int,%20int,%20boolean,%20int)"><code>monteCarloP(double, int, int, boolean, int)</code></a>.
Deprecated as of version 3.6, as this method is no longer needed.</div>
<dl><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../constant-values.html#org.apache.commons.math3.stat.inference.KolmogorovSmirnovTest.MONTE_CARLO_ITERATIONS">Constant Field Values</a></dd></dl>
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<pre>public KolmogorovSmirnovTest()</pre>
<div class="block">Construct a KolmogorovSmirnovTest instance with a default random data generator.</div>
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<pre><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Deprecated.html?is-external=true" title="class or interface in java.lang">@Deprecated</a>
public KolmogorovSmirnovTest(<a href="../../../../../../org/apache/commons/math3/random/RandomGenerator.html" title="interface in org.apache.commons.math3.random">RandomGenerator</a> rng)</pre>
<div class="block"><span class="strong">Deprecated.</span> </div>
<div class="block">Construct a KolmogorovSmirnovTest with the provided random data generator.
The #monteCarloP(double, int, int, boolean, int) that uses the generator supplied to this
constructor is deprecated as of version 3.6.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>rng</code> - random data generator used by <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#monteCarloP(double,%20int,%20int,%20boolean,%20int)"><code>monteCarloP(double, int, int, boolean, int)</code></a></dd></dl>
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<pre>public double kolmogorovSmirnovTest(<a href="../../../../../../org/apache/commons/math3/distribution/RealDistribution.html" title="interface in org.apache.commons.math3.distribution">RealDistribution</a> distribution,
double[] data,
boolean exact)</pre>
<div class="block">Computes the <i>p-value</i>, or <i>observed significance level</i>, of a one-sample <a
href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>data</code> conforms to <code>distribution</code>. If
<code>exact</code> is true, the distribution used to compute the p-value is computed using
extended precision. See <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#cdfExact(double,%20int)"><code>cdfExact(double, int)</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>distribution</code> - reference distribution</dd><dd><code>data</code> - sample being being evaluated</dd><dd><code>exact</code> - whether or not to force exact computation of the p-value</dd>
<dt><span class="strong">Returns:</span></dt><dd>the p-value associated with the null hypothesis that <code>data</code> is a sample from
<code>distribution</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/InsufficientDataException.html" title="class in org.apache.commons.math3.exception">InsufficientDataException</a></code> - if <code>data</code> does not have length at least 2</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NullArgumentException.html" title="class in org.apache.commons.math3.exception">NullArgumentException</a></code> - if <code>data</code> is null</dd></dl>
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<h4>kolmogorovSmirnovStatistic</h4>
<pre>public double kolmogorovSmirnovStatistic(<a href="../../../../../../org/apache/commons/math3/distribution/RealDistribution.html" title="interface in org.apache.commons.math3.distribution">RealDistribution</a> distribution,
double[] data)</pre>
<div class="block">Computes the one-sample Kolmogorov-Smirnov test statistic, \(D_n=\sup_x |F_n(x)-F(x)|\) where
\(F\) is the distribution (cdf) function associated with <code>distribution</code>, \(n\) is the
length of <code>data</code> and \(F_n\) is the empirical distribution that puts mass \(1/n\) at
each of the values in <code>data</code>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>distribution</code> - reference distribution</dd><dd><code>data</code> - sample being evaluated</dd>
<dt><span class="strong">Returns:</span></dt><dd>Kolmogorov-Smirnov statistic \(D_n\)</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/InsufficientDataException.html" title="class in org.apache.commons.math3.exception">InsufficientDataException</a></code> - if <code>data</code> does not have length at least 2</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NullArgumentException.html" title="class in org.apache.commons.math3.exception">NullArgumentException</a></code> - if <code>data</code> is null</dd></dl>
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<pre>public double kolmogorovSmirnovTest(double[] x,
double[] y,
boolean strict)</pre>
<div class="block">Computes the <i>p-value</i>, or <i>observed significance level</i>, of a two-sample <a
href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>x</code> and <code>y</code> are samples drawn from the same
probability distribution. Specifically, what is returned is an estimate of the probability
that the <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovStatistic(double[],%20double[])"><code>kolmogorovSmirnovStatistic(double[], double[])</code></a> associated with a randomly
selected partition of the combined sample into subsamples of sizes <code>x.length</code> and
<code>y.length</code> will strictly exceed (if <code>strict</code> is <code>true</code>) or be at least as
large as <code>strict = false</code>) as <code>kolmogorovSmirnovStatistic(x, y)</code>.
<ul>
<li>For small samples (where the product of the sample sizes is less than
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#LARGE_SAMPLE_PRODUCT">10000</a>), the exact p-value is computed using the method presented
in [4], implemented in <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#exactP(double,%20int,%20int,%20boolean)"><code>exactP(double, int, int, boolean)</code></a>. </li>
<li>When the product of the sample sizes exceeds <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#LARGE_SAMPLE_PRODUCT">10000</a>, the
asymptotic distribution of \(D_{n,m}\) is used. See <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#approximateP(double,%20int,%20int)"><code>approximateP(double, int, int)</code></a>
for details on the approximation.</li>
</ul><p>
If <code>x.length * y.length</code> < <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#LARGE_SAMPLE_PRODUCT">10000</a> and the combined set of values in
<code>x</code> and <code>y</code> contains ties, random jitter is added to <code>x</code> and <code>y</code> to
break ties before computing \(D_{n,m}\) and the p-value. The jitter is uniformly distributed
on (-minDelta / 2, minDelta / 2) where minDelta is the smallest pairwise difference between
values in the combined sample.</p>
<p>
If ties are known to be present in the data, <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#bootstrap(double[],%20double[],%20int,%20boolean)"><code>bootstrap(double[], double[], int, boolean)</code></a>
may be used as an alternative method for estimating the p-value.</p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first sample dataset</dd><dd><code>y</code> - second sample dataset</dd><dd><code>strict</code> - whether or not the probability to compute is expressed as a strict inequality
(ignored for large samples)</dd>
<dt><span class="strong">Returns:</span></dt><dd>p-value associated with the null hypothesis that <code>x</code> and <code>y</code> represent
samples from the same distribution</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/InsufficientDataException.html" title="class in org.apache.commons.math3.exception">InsufficientDataException</a></code> - if either <code>x</code> or <code>y</code> does not have length at
least 2</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NullArgumentException.html" title="class in org.apache.commons.math3.exception">NullArgumentException</a></code> - if either <code>x</code> or <code>y</code> is null</dd><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#bootstrap(double[],%20double[],%20int,%20boolean)"><code>bootstrap(double[], double[], int, boolean)</code></a></dd></dl>
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<pre>public double kolmogorovSmirnovTest(double[] x,
double[] y)</pre>
<div class="block">Computes the <i>p-value</i>, or <i>observed significance level</i>, of a two-sample <a
href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>x</code> and <code>y</code> are samples drawn from the same
probability distribution. Assumes the strict form of the inequality used to compute the
p-value. See <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovTest(org.apache.commons.math3.distribution.RealDistribution,%20double[],%20boolean)"><code>kolmogorovSmirnovTest(RealDistribution, double[], boolean)</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first sample dataset</dd><dd><code>y</code> - second sample dataset</dd>
<dt><span class="strong">Returns:</span></dt><dd>p-value associated with the null hypothesis that <code>x</code> and <code>y</code> represent
samples from the same distribution</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/InsufficientDataException.html" title="class in org.apache.commons.math3.exception">InsufficientDataException</a></code> - if either <code>x</code> or <code>y</code> does not have length at
least 2</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NullArgumentException.html" title="class in org.apache.commons.math3.exception">NullArgumentException</a></code> - if either <code>x</code> or <code>y</code> is null</dd></dl>
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<pre>public double kolmogorovSmirnovStatistic(double[] x,
double[] y)</pre>
<div class="block">Computes the two-sample Kolmogorov-Smirnov test statistic, \(D_{n,m}=\sup_x |F_n(x)-F_m(x)|\)
where \(n\) is the length of <code>x</code>, \(m\) is the length of <code>y</code>, \(F_n\) is the
empirical distribution that puts mass \(1/n\) at each of the values in <code>x</code> and \(F_m\)
is the empirical distribution of the <code>y</code> values.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first sample</dd><dd><code>y</code> - second sample</dd>
<dt><span class="strong">Returns:</span></dt><dd>test statistic \(D_{n,m}\) used to evaluate the null hypothesis that <code>x</code> and
<code>y</code> represent samples from the same underlying distribution</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/InsufficientDataException.html" title="class in org.apache.commons.math3.exception">InsufficientDataException</a></code> - if either <code>x</code> or <code>y</code> does not have length at
least 2</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NullArgumentException.html" title="class in org.apache.commons.math3.exception">NullArgumentException</a></code> - if either <code>x</code> or <code>y</code> is null</dd></dl>
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<pre>public double kolmogorovSmirnovTest(<a href="../../../../../../org/apache/commons/math3/distribution/RealDistribution.html" title="interface in org.apache.commons.math3.distribution">RealDistribution</a> distribution,
double[] data)</pre>
<div class="block">Computes the <i>p-value</i>, or <i>observed significance level</i>, of a one-sample <a
href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>data</code> conforms to <code>distribution</code>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>distribution</code> - reference distribution</dd><dd><code>data</code> - sample being being evaluated</dd>
<dt><span class="strong">Returns:</span></dt><dd>the p-value associated with the null hypothesis that <code>data</code> is a sample from
<code>distribution</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/InsufficientDataException.html" title="class in org.apache.commons.math3.exception">InsufficientDataException</a></code> - if <code>data</code> does not have length at least 2</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NullArgumentException.html" title="class in org.apache.commons.math3.exception">NullArgumentException</a></code> - if <code>data</code> is null</dd></dl>
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<pre>public boolean kolmogorovSmirnovTest(<a href="../../../../../../org/apache/commons/math3/distribution/RealDistribution.html" title="interface in org.apache.commons.math3.distribution">RealDistribution</a> distribution,
double[] data,
double alpha)</pre>
<div class="block">Performs a <a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov
test</a> evaluating the null hypothesis that <code>data</code> conforms to <code>distribution</code>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>distribution</code> - reference distribution</dd><dd><code>data</code> - sample being being evaluated</dd><dd><code>alpha</code> - significance level of the test</dd>
<dt><span class="strong">Returns:</span></dt><dd>true iff the null hypothesis that <code>data</code> is a sample from <code>distribution</code>
can be rejected with confidence 1 - <code>alpha</code></dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/InsufficientDataException.html" title="class in org.apache.commons.math3.exception">InsufficientDataException</a></code> - if <code>data</code> does not have length at least 2</dd>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/NullArgumentException.html" title="class in org.apache.commons.math3.exception">NullArgumentException</a></code> - if <code>data</code> is null</dd></dl>
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<pre>public double bootstrap(double[] x,
double[] y,
int iterations,
boolean strict)</pre>
<div class="block">Estimates the <i>p-value</i> of a two-sample
<a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test"> Kolmogorov-Smirnov test</a>
evaluating the null hypothesis that <code>x</code> and <code>y</code> are samples drawn from the same
probability distribution. This method estimates the p-value by repeatedly sampling sets of size
<code>x.length</code> and <code>y.length</code> from the empirical distribution of the combined sample.
When <code>strict</code> is true, this is equivalent to the algorithm implemented in the R function
<code>ks.boot</code>, described in <pre>
Jasjeet S. Sekhon. 2011. 'Multivariate and Propensity Score Matching
Software with Automated Balance Optimization: The Matching package for R.'
Journal of Statistical Software, 42(7): 1-52.
</pre></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first sample</dd><dd><code>y</code> - second sample</dd><dd><code>iterations</code> - number of bootstrap resampling iterations</dd><dd><code>strict</code> - whether or not the null hypothesis is expressed as a strict inequality</dd>
<dt><span class="strong">Returns:</span></dt><dd>estimated p-value</dd></dl>
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<pre>public double bootstrap(double[] x,
double[] y,
int iterations)</pre>
<div class="block">Computes <code>bootstrap(x, y, iterations, true)</code>.
This is equivalent to ks.boot(x,y, nboots=iterations) using the R Matching
package function. See #bootstrap(double[], double[], int, boolean).</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first sample</dd><dd><code>y</code> - second sample</dd><dd><code>iterations</code> - number of bootstrap resampling iterations</dd>
<dt><span class="strong">Returns:</span></dt><dd>estimated p-value</dd></dl>
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<pre>public double cdf(double d,
int n)
throws <a href="../../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></pre>
<div class="block">Calculates \(P(D_n < d)\) using the method described in [1] with quick decisions for extreme
values given in [2] (see above). The result is not exact as with
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#cdfExact(double,%20int)"><code>cdfExact(double, int)</code></a> because calculations are based on
<code>double</code> rather than <a href="../../../../../../org/apache/commons/math3/fraction/BigFraction.html" title="class in org.apache.commons.math3.fraction"><code>BigFraction</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>d</code> - statistic</dd><dd><code>n</code> - sample size</dd>
<dt><span class="strong">Returns:</span></dt><dd>\(P(D_n < d)\)</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if algorithm fails to convert <code>h</code> to a
<a href="../../../../../../org/apache/commons/math3/fraction/BigFraction.html" title="class in org.apache.commons.math3.fraction"><code>BigFraction</code></a> in expressing <code>d</code> as \((k
- h) / m\) for integer <code>k, m</code> and \(0 \le h < 1\)</dd></dl>
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<pre>public double cdfExact(double d,
int n)
throws <a href="../../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></pre>
<div class="block">Calculates <code>P(D_n < d)</code>. The result is exact in the sense that BigFraction/BigReal is
used everywhere at the expense of very slow execution time. Almost never choose this in real
applications unless you are very sure; this is almost solely for verification purposes.
Normally, you would choose <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#cdf(double,%20int)"><code>cdf(double, int)</code></a>. See the class
javadoc for definitions and algorithm description.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>d</code> - statistic</dd><dd><code>n</code> - sample size</dd>
<dt><span class="strong">Returns:</span></dt><dd>\(P(D_n < d)\)</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if the algorithm fails to convert <code>h</code> to a
<a href="../../../../../../org/apache/commons/math3/fraction/BigFraction.html" title="class in org.apache.commons.math3.fraction"><code>BigFraction</code></a> in expressing <code>d</code> as \((k
- h) / m\) for integer <code>k, m</code> and \(0 \le h < 1\)</dd></dl>
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<pre>public double cdf(double d,
int n,
boolean exact)
throws <a href="../../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></pre>
<div class="block">Calculates <code>P(D_n < d)</code> using method described in [1] with quick decisions for extreme
values given in [2] (see above).</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>d</code> - statistic</dd><dd><code>n</code> - sample size</dd><dd><code>exact</code> - whether the probability should be calculated exact using
<a href="../../../../../../org/apache/commons/math3/fraction/BigFraction.html" title="class in org.apache.commons.math3.fraction"><code>BigFraction</code></a> everywhere at the expense of
very slow execution time, or if <code>double</code> should be used convenient places to
gain speed. Almost never choose <code>true</code> in real applications unless you are very
sure; <code>true</code> is almost solely for verification purposes.</dd>
<dt><span class="strong">Returns:</span></dt><dd>\(P(D_n < d)\)</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if algorithm fails to convert <code>h</code> to a
<a href="../../../../../../org/apache/commons/math3/fraction/BigFraction.html" title="class in org.apache.commons.math3.fraction"><code>BigFraction</code></a> in expressing <code>d</code> as \((k
- h) / m\) for integer <code>k, m</code> and \(0 \le h < 1\).</dd></dl>
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<h4>pelzGood</h4>
<pre>public double pelzGood(double d,
int n)</pre>
<div class="block">Computes the Pelz-Good approximation for \(P(D_n < d)\) as described in [2] in the class javadoc.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>d</code> - value of d-statistic (x in [2])</dd><dd><code>n</code> - sample size</dd>
<dt><span class="strong">Returns:</span></dt><dd>\(P(D_n < d)\)</dd><dt><span class="strong">Since:</span></dt>
<dd>3.4</dd></dl>
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<pre>public double ksSum(double t,
double tolerance,
int maxIterations)</pre>
<div class="block">Computes \( 1 + 2 \sum_{i=1}^\infty (-1)^i e^{-2 i^2 t^2} \) stopping when successive partial
sums are within <code>tolerance</code> of one another, or when <code>maxIterations</code> partial sums
have been computed. If the sum does not converge before <code>maxIterations</code> iterations a
<a href="../../../../../../org/apache/commons/math3/exception/TooManyIterationsException.html" title="class in org.apache.commons.math3.exception"><code>TooManyIterationsException</code></a> is thrown.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>t</code> - argument</dd><dd><code>tolerance</code> - Cauchy criterion for partial sums</dd><dd><code>maxIterations</code> - maximum number of partial sums to compute</dd>
<dt><span class="strong">Returns:</span></dt><dd>Kolmogorov sum evaluated at t</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../../org/apache/commons/math3/exception/TooManyIterationsException.html" title="class in org.apache.commons.math3.exception">TooManyIterationsException</a></code> - if the series does not converge</dd></dl>
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<pre>public double exactP(double d,
int n,
int m,
boolean strict)</pre>
<div class="block">Computes \(P(D_{n,m} > d)\) if <code>strict</code> is <code>true</code>; otherwise \(P(D_{n,m} \ge
d)\), where \(D_{n,m}\) is the 2-sample Kolmogorov-Smirnov statistic. See
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovStatistic(double[],%20double[])"><code>kolmogorovSmirnovStatistic(double[], double[])</code></a> for the definition of \(D_{n,m}\).
<p>
The returned probability is exact, implemented by unwinding the recursive function
definitions presented in [4] (class javadoc).
</p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>d</code> - D-statistic value</dd><dd><code>n</code> - first sample size</dd><dd><code>m</code> - second sample size</dd><dd><code>strict</code> - whether or not the probability to compute is expressed as a strict inequality</dd>
<dt><span class="strong">Returns:</span></dt><dd>probability that a randomly selected m-n partition of m + n generates \(D_{n,m}\)
greater than (resp. greater than or equal to) <code>d</code></dd></dl>
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<pre>public double approximateP(double d,
int n,
int m)</pre>
<div class="block">Uses the Kolmogorov-Smirnov distribution to approximate \(P(D_{n,m} > d)\) where \(D_{n,m}\)
is the 2-sample Kolmogorov-Smirnov statistic. See
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovStatistic(double[],%20double[])"><code>kolmogorovSmirnovStatistic(double[], double[])</code></a> for the definition of \(D_{n,m}\).
<p>
Specifically, what is returned is \(1 - k(d \sqrt{mn / (m + n)})\) where \(k(t) = 1 + 2
\sum_{i=1}^\infty (-1)^i e^{-2 i^2 t^2}\). See <a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#ksSum(double,%20double,%20int)"><code>ksSum(double, double, int)</code></a> for
details on how convergence of the sum is determined. This implementation passes <code>ksSum</code>
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#KS_SUM_CAUCHY_CRITERION">1.0E-20</a> as <code>tolerance</code> and
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#MAXIMUM_PARTIAL_SUM_COUNT">100000</a> as <code>maxIterations</code>.
</p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>d</code> - D-statistic value</dd><dd><code>n</code> - first sample size</dd><dd><code>m</code> - second sample size</dd>
<dt><span class="strong">Returns:</span></dt><dd>approximate probability that a randomly selected m-n partition of m + n generates
\(D_{n,m}\) greater than <code>d</code></dd></dl>
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<pre>public double monteCarloP(double d,
int n,
int m,
boolean strict,
int iterations)</pre>
<div class="block">Uses Monte Carlo simulation to approximate \(P(D_{n,m} > d)\) where \(D_{n,m}\) is the
2-sample Kolmogorov-Smirnov statistic. See
<a href="../../../../../../org/apache/commons/math3/stat/inference/KolmogorovSmirnovTest.html#kolmogorovSmirnovStatistic(double[],%20double[])"><code>kolmogorovSmirnovStatistic(double[], double[])</code></a> for the definition of \(D_{n,m}\).
<p>
The simulation generates <code>iterations</code> random partitions of <code>m + n</code> into an
<code>n</code> set and an <code>m</code> set, computing \(D_{n,m}\) for each partition and returning
the proportion of values that are greater than <code>d</code>, or greater than or equal to
<code>d</code> if <code>strict</code> is <code>false</code>.
</p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>d</code> - D-statistic value</dd><dd><code>n</code> - first sample size</dd><dd><code>m</code> - second sample size</dd><dd><code>iterations</code> - number of random partitions to generate</dd><dd><code>strict</code> - whether or not the probability to compute is expressed as a strict inequality</dd>
<dt><span class="strong">Returns:</span></dt><dd>proportion of randomly generated m-n partitions of m + n that result in \(D_{n,m}\)
greater than (resp. greater than or equal to) <code>d</code></dd></dl>
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