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* Kolmogorov-Smirnov distribution #421 Add a new distribution, kolmogorov_smirnov_distribution, which takes a parameter that represents the number of observations used in a Kolmogorov-Smirnov test. (The K-S test is a popular test for comparing two CDFs, but the test statistic is not implemented here.) This implementation includes Kolmogorov's original 1st order Taylor expansion. There is a literature on the distribution's other mathematical properties (higher order terms and exact version); this literature is summarized in the main header file for anyone who may want to expand the implementation later. The CDF is implemented using a Jacobi theta function, and the PDF is a hand-rolled derivative of that function. Quantiles plug the CDF and PDF into a Newton-Raphson iteration. The mean and variance have nice closed-form expressions, and the mode uses a dumb run-time maximizer. This commit includes graphs, a ULP plotter for the PDF, and the usual compilation and numerical tests. The test file is on the small side, but it integrates the distribution from zero to infinity, and covers the quantiles pretty well. As of now the numerical tests only verify self-consistency (e.g. distribution moments and CDF-quantile relations), so there's room to add some external checks. * Implement skewness for K-S distribution [CI SKIP] The third moment integrates nicely with the help of Apery's constant (zeta_three). Verify the result via quadrature. * Implement kurtosis for the K-S distribution Verify the result via quadrature.
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227 lines
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<div class="chapter">
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<div class="titlepage"><div><div><h1 class="title">
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<a name="dist"></a>Chapter 5. Statistical Distributions and Functions</h1></div></div></div>
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<div class="toc">
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<p><b>Table of Contents</b></p>
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<dl class="toc">
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<dt><span class="section"><a href="math_toolkit/stat_tut.html">Statistical Distributions Tutorial</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/stat_tut/overview.html">Overview of Statistical
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Distributions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/stat_tut/overview/headers.html">Headers and
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Namespaces</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/overview/objects.html">Distributions
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are Objects</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/overview/generic.html">Generic operations
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common to all distributions are non-member functions</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/overview/complements.html">Complements
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are supported too - and when to use them</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/overview/parameters.html">Parameters
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can be calculated</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/overview/summary.html">Summary</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg.html">Worked Examples</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/dist_construct_eg.html">Distribution
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Construction Examples</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/st_eg.html">Student's t Distribution
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Examples</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/st_eg/tut_mean_intervals.html">Calculating
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confidence intervals on the mean with the Students-t distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/st_eg/tut_mean_test.html">Testing
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a sample mean for difference from a "true" mean</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/st_eg/tut_mean_size.html">Estimating
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how large a sample size would have to become in order to give a significant
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Students-t test result with a single sample test</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/st_eg/two_sample_students_t.html">Comparing
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the means of two samples with the Students-t test</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/st_eg/paired_st.html">Comparing
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two paired samples with the Student's t distribution</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/cs_eg.html">Chi Squared Distribution
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Examples</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/cs_eg/chi_sq_intervals.html">Confidence
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Intervals on the Standard Deviation</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/cs_eg/chi_sq_test.html">Chi-Square
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Test for the Standard Deviation</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/cs_eg/chi_sq_size.html">Estimating
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the Required Sample Sizes for a Chi-Square Test for the Standard Deviation</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/f_eg.html">F Distribution Examples</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/binom_eg.html">Binomial Distribution
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Examples</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/binom_eg/binomial_coinflip_example.html">Binomial
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Coin-Flipping Example</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/binom_eg/binomial_quiz_example.html">Binomial
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Quiz Example</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/binom_eg/binom_conf.html">Calculating
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Confidence Limits on the Frequency of Occurrence for a Binomial Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/binom_eg/binom_size_eg.html">Estimating
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Sample Sizes for a Binomial Distribution.</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/geometric_eg.html">Geometric Distribution
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Examples</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/neg_binom_eg.html">Negative Binomial
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Distribution Examples</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/neg_binom_eg/neg_binom_conf.html">Calculating
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Confidence Limits on the Frequency of Occurrence for the Negative Binomial
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html">Estimating
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Sample Sizes for the Negative Binomial.</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/neg_binom_eg/negative_binomial_example1.html">Negative
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Binomial Sales Quota Example.</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/neg_binom_eg/negative_binomial_example2.html">Negative
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Binomial Table Printing Example.</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/normal_example.html">Normal Distribution
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Examples</a></span></dt>
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<dd><dl><dt><span class="section"><a href="math_toolkit/stat_tut/weg/normal_example/normal_misc.html">Some
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Miscellaneous Examples of the Normal (Gaussian) Distribution</a></span></dt></dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/inverse_chi_squared_eg.html">Inverse
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Chi-Squared Distribution Bayes Example</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/nccs_eg.html">Non Central Chi
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Squared Example</a></span></dt>
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<dd><dl><dt><span class="section"><a href="math_toolkit/stat_tut/weg/nccs_eg/nccs_power_eg.html">Tables
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of the power function of the chi<sup>2</sup> test.</a></span></dt></dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/error_eg.html">Error Handling
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Example</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/find_eg.html">Find Location and
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Scale Examples</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/find_eg/find_location_eg.html">Find
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Location (Mean) Example</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/find_eg/find_scale_eg.html">Find
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Scale (Standard Deviation) Example</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/find_eg/find_mean_and_sd_eg.html">Find
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mean and standard deviation example</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/nag_library.html">Comparison with
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C, R, FORTRAN-style Free Functions</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/weg/c_sharp.html">Using the Distributions
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from Within C#</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/stat_tut/variates.html">Random Variates and Distribution
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Parameters</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/stat_tut/dist_params.html">Discrete Probability
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Distributions</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/dist_ref.html">Statistical Distributions Reference</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/dist_ref/nmp.html">Non-Member Properties</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists.html">Distributions</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/arcine_dist.html">Arcsine Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/bernoulli_dist.html">Bernoulli
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/beta_dist.html">Beta Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/binomial_dist.html">Binomial
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/cauchy_dist.html">Cauchy-Lorentz
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/chi_squared_dist.html">Chi Squared
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/empirical_cdf.html">Empirical
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Cumulative Distribution Function</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/exp_dist.html">Exponential Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/extreme_dist.html">Extreme Value
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/f_dist.html">F Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/gamma_dist.html">Gamma (and
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Erlang) Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/geometric_dist.html">Geometric
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/hyperexponential_dist.html">Hyperexponential
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/hypergeometric_dist.html">Hypergeometric
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/inverse_chi_squared_dist.html">Inverse
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Chi Squared Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/inverse_gamma_dist.html">Inverse
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Gamma Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/inverse_gaussian_dist.html">Inverse
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Gaussian (or Inverse Normal) Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/kolmogorov_smirnov_dist.html">Kolmogorov-Smirnov
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/laplace_dist.html">Laplace Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/logistic_dist.html">Logistic
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/lognormal_dist.html">Log Normal
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/negative_binomial_dist.html">Negative
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Binomial Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/nc_beta_dist.html">Noncentral
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Beta Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/nc_chi_squared_dist.html">Noncentral
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Chi-Squared Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/nc_f_dist.html">Noncentral F
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/nc_t_dist.html">Noncentral T
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/normal_dist.html">Normal (Gaussian)
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/pareto.html">Pareto Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/poisson_dist.html">Poisson Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/rayleigh.html">Rayleigh Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/skew_normal_dist.html">Skew
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Normal Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/students_t_dist.html">Students
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t Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/triangular_dist.html">Triangular
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Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/uniform_dist.html">Uniform Distribution</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dists/weibull_dist.html">Weibull Distribution</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/dist_ref/dist_algorithms.html">Distribution Algorithms</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/future.html">Extras/Future Directions</a></span></dt>
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</dl>
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</div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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</p>
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</div></td>
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