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 // Kolmogorov-Smirnov 1st order asymptotic distribution
 // Copyright Evan Miller 2020
 //
 // Use, modification and distribution are subject to the
 // Boost Software License, Version 1.0. (See accompanying file
 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 //
 // The Kolmogorov-Smirnov test in statistics compares two empirical distributions,
 // or an empirical distribution against any theoretical distribution. It makes
 // use of a specific distribution which doesn't have a formal name, but which
 // is often called the Kolmogorv-Smirnov distribution for lack of anything
 // better. This file implements the limiting form of this distribution, first
 // identified by Andrey Kolmogorov in
 //
 // Kolmogorov, A. (1933) "Sulla Determinazione Empirica di una Legge di
 // Distribuzione." Giornale dell' Istituto Italiano degli Attuari
 //
 // This limiting form of the CDF is a first-order Taylor expansion that is
 // easily implemented by the fourth Jacobi Theta function (setting z=0). The
 // PDF is then implemented here as a derivative of the Theta function. Note
 // that this derivative is with respect to x, which enters into \tau, and not
 // with respect to the z argument, which is always zero, and so the derivative
 // identities in DLMF 20.4 do not apply here.
 //
 // A higher order order expansion is possible, and was first outlined by
 //
 // Pelz W, Good IJ (1976). "Approximating the Lower Tail-Areas of the
 // Kolmogorov-Smirnov One-sample Statistic." Journal of the Royal Statistical
 // Society B.
 //
 // The terms in this expansion get fairly complicated, and as far as I know the
 // Pelz-Good expansion is not used in any statistics software. Someone could
 // consider updating this implementation to use the Pelz-Good expansion in the
 // future, but the math gets considerably hairier with each additional term.
 //
 // A formula for an exact version of the Kolmogorov-Smirnov test is laid out in
 // Equation 2.4.4 of
 //
 // Durbin J (1973). "Distribution Theory for Tests Based on the Sample
 // Distribution Func- tion." In SIAM CBMS-NSF Regional Conference Series in
 // Applied Mathematics. SIAM, Philadelphia, PA.
 //
 // which is available in book form from Amazon and others. This exact version
 // involves taking powers of large matrices. To do that right you need to
 // compute eigenvalues and eigenvectors, which are beyond the scope of Boost.
 // (Some recent work indicates the exact form can also be computed via FFT, see
 // https://cran.r-project.org/web/packages/KSgeneral/KSgeneral.pdf).
 //
 // Even if the CDF of the exact distribution could be computed using Boost
 // libraries (which would be cumbersome), the PDF would present another
 // difficulty. Therefore I am limiting this implementation to the asymptotic
 // form, even though the exact form has trivial values for certain specific
 // values of x and n. For more on trivial values see
 //
 // Ruben H, Gambino J (1982). "The Exact Distribution of Kolmogorov's Statistic
 // Dn for n <= 10." Annals of the Institute of Statistical Mathematics.
 // 
 // For a good bibliography and overview of the various algorithms, including
 // both exact and asymptotic forms, see
 // https://www.jstatsoft.org/article/view/v039i11
 //
 // As for this implementation: the distribution is parameterized by n (number
 // of observations) in the spirit of chi-squared's degrees of freedom. It then
 // takes a single argument x. In terms of the Kolmogorov-Smirnov statistical
 // test, x represents the distribution of D_n, where D_n is the maximum
 // difference between the CDFs being compared, that is,
 //
 //   D_n = sup|F_n(x) - G(x)|
 //
 // In the exact distribution, x is confined to the support [0, 1], but in this
 // limiting approximation, we allow x to exceed unity (similar to how a normal
 // approximation always spills over any boundaries).
 //
 // As mentioned previously, the CDF is implemented using the \tau
 // parameterization of the fourth Jacobi Theta function as
 //
 // CDF=theta_4(0|2*x*x*n/pi)
 //
 // The PDF is a hand-coded derivative of that function. Actually, there are two
 // (independent) derivatives, as separate code paths are used for "small x"
 // (2*x*x*n < pi) and "large x", mirroring the separate code paths in the
 // Jacobi Theta implementation to achieve fast convergence. Quantiles are
 // computed using a Newton-Raphson iteration from an initial guess that I
 // arrived at by trial and error.
 //
 // The mean and variance are implemented using simple closed-form expressions.
 // Skewness and kurtosis use slightly more complicated closed-form expressions
 // that involve the zeta function. The mode is calculated at run-time by
 // maximizing the PDF. If you have an analytical solution for the mode, feel
 // free to plop it in.
 //
 // The CDF and PDF could almost certainly be re-implemented and sped up using a
 // polynomial or rational approximation, since the only meaningful argument is
 // x * sqrt(n). But that is left as an exercise for the next maintainer.
 //
 // In the future, the Pelz-Good approximation could be added. I suggest adding
 // a second parameter representing the order, e.g.
 //
 // kolmogorov_smirnov_dist<>(100) // N=100, order=1
 // kolmogorov_smirnov_dist<>(100, 1) // N=100, order=1, i.e. Kolmogorov's formula
 // kolmogorov_smirnov_dist<>(100, 4) // N=100, order=4, i.e. Pelz-Good formula
 //
 // The exact distribution could be added to the API with a special order
 // parameter (e.g. 0 or infinity), or a separate distribution type altogether
 // (e.g. kolmogorov_smirnov_exact_distribution).
 //
 #ifndef BOOST_MATH_DISTRIBUTIONS_KOLMOGOROV_SMIRNOV_HPP
 #define BOOST_MATH_DISTRIBUTIONS_KOLMOGOROV_SMIRNOV_HPP
 
 #include <boost/math/distributions/fwd.hpp>
 #include <boost/math/distributions/complement.hpp>
 #include <boost/math/distributions/detail/common_error_handling.hpp>
 #include <boost/math/special_functions/jacobi_theta.hpp>
 #include <boost/math/tools/tuple.hpp>
 #include <boost/math/tools/roots.hpp> // Newton-Raphson
 #include <boost/math/tools/minima.hpp> // For the mode
 
 namespace boost { namespace math {
 
 namespace detail {
 template <class RealType>
 inline RealType kolmogorov_smirnov_quantile_guess(RealType p) {
     // Choose a starting point for the Newton-Raphson iteration
     if (p > 0.9)
         return RealType(1.8) - 5 * (1 - p);
     if (p < 0.3)
         return p + RealType(0.45);
     return p + RealType(0.3);
 }
 
 // d/dk (theta2(0, 1/(2*k*k/M_PI))/sqrt(2*k*k*M_PI))
 template <class RealType, class Policy>
 RealType kolmogorov_smirnov_pdf_small_x(RealType x, RealType n, const Policy&) {
     BOOST_MATH_STD_USING
     RealType value = RealType(0), delta = RealType(0), last_delta = RealType(0);
     RealType eps = policies::get_epsilon<RealType, Policy>();
     int i = 0;
     RealType pi2 = constants::pi_sqr<RealType>();
     RealType x2n = x*x*n;
     if (x2n*x2n == 0.0) {
         return static_cast<RealType>(0);
     }
     while (true) {
         delta = exp(-RealType(i+0.5)*RealType(i+0.5)*pi2/(2*x2n)) * (RealType(i+0.5)*RealType(i+0.5)*pi2 - x2n);
 
         if (delta == 0.0)
             break;
 
         if (last_delta != 0.0 && fabs(delta/last_delta) < eps)
             break;
 
         value += delta + delta;
         last_delta = delta;
         i++;
     }
 
     return value * sqrt(n) * constants::root_half_pi<RealType>() / (x2n*x2n);
 }
 
 // d/dx (theta4(0, 2*x*x*n/M_PI))
 template <class RealType, class Policy>
 inline RealType kolmogorov_smirnov_pdf_large_x(RealType x, RealType n, const Policy&) {
     BOOST_MATH_STD_USING
     RealType value = RealType(0), delta = RealType(0), last_delta = RealType(0);
     RealType eps = policies::get_epsilon<RealType, Policy>();
     int i = 1;
     while (true) {
         delta = 8*x*i*i*exp(-2*i*i*x*x*n);
 
         if (delta == 0.0)
             break;
 
         if (last_delta != 0.0 && fabs(delta / last_delta) < eps)
             break;
 
         if (i%2 == 0)
             delta = -delta;
 
         value += delta;
         last_delta = delta;
         i++;
     }
 
     return value * n;
 }
 
 } // detail
 
 template <class RealType = double, class Policy = policies::policy<> >
     class kolmogorov_smirnov_distribution
 {
     public:
         typedef RealType value_type;
         typedef Policy policy_type;
 
         // Constructor
     kolmogorov_smirnov_distribution( RealType n ) : n_obs_(n)
     {
         RealType result;
         detail::check_df(
                 "boost::math::kolmogorov_smirnov_distribution<%1%>::kolmogorov_smirnov_distribution", n_obs_, &result, Policy());
     }
 
     RealType number_of_observations()const
     {
         return n_obs_;
     }
 
     private:
 
     RealType n_obs_; // positive integer
 };
 
 typedef kolmogorov_smirnov_distribution<double> kolmogorov_k; // Convenience typedef for double version.
 
 #ifdef __cpp_deduction_guides
 template <class RealType>
 kolmogorov_smirnov_distribution(RealType)->kolmogorov_smirnov_distribution<typename boost::math::tools::promote_args<RealType>::type>;
 #endif
 
 namespace detail {
 template <class RealType, class Policy>
 struct kolmogorov_smirnov_quantile_functor
 {
   kolmogorov_smirnov_quantile_functor(const boost::math::kolmogorov_smirnov_distribution<RealType, Policy> dist, RealType const& p)
     : distribution(dist), prob(p)
   {
   }
 
   boost::math::tuple<RealType, RealType> operator()(RealType const& x)
   {
     RealType fx = cdf(distribution, x) - prob;  // Difference cdf - value - to minimize.
     RealType dx = pdf(distribution, x); // pdf is 1st derivative.
     // return both function evaluation difference f(x) and 1st derivative f'(x).
     return boost::math::make_tuple(fx, dx);
   }
 private:
   const boost::math::kolmogorov_smirnov_distribution<RealType, Policy> distribution;
   RealType prob;
 };
 
 template <class RealType, class Policy>
 struct kolmogorov_smirnov_complementary_quantile_functor
 {
   kolmogorov_smirnov_complementary_quantile_functor(const boost::math::kolmogorov_smirnov_distribution<RealType, Policy> dist, RealType const& p)
     : distribution(dist), prob(p)
   {
   }
 
   boost::math::tuple<RealType, RealType> operator()(RealType const& x)
   {
     RealType fx = cdf(complement(distribution, x)) - prob;  // Difference cdf - value - to minimize.
     RealType dx = -pdf(distribution, x); // pdf is the negative of the derivative of (1-CDF)
     // return both function evaluation difference f(x) and 1st derivative f'(x).
     return boost::math::make_tuple(fx, dx);
   }
 private:
   const boost::math::kolmogorov_smirnov_distribution<RealType, Policy> distribution;
   RealType prob;
 };
 
 template <class RealType, class Policy>
 struct kolmogorov_smirnov_negative_pdf_functor
 {
     RealType operator()(RealType const& x) {
         if (2*x*x < constants::pi<RealType>()) {
             return -kolmogorov_smirnov_pdf_small_x(x, static_cast<RealType>(1), Policy());
         }
         return -kolmogorov_smirnov_pdf_large_x(x, static_cast<RealType>(1), Policy());
     }
 };
 } // namespace detail
 
 template <class RealType, class Policy>
 inline const std::pair<RealType, RealType> range(const kolmogorov_smirnov_distribution<RealType, Policy>& /*dist*/)
 { // Range of permissible values for random variable x.
    using boost::math::tools::max_value;
    return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>());
 }
 
 template <class RealType, class Policy>
 inline const std::pair<RealType, RealType> support(const kolmogorov_smirnov_distribution<RealType, Policy>& /*dist*/)
 { // Range of supported values for random variable x.
    // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
    // In the exact distribution, the upper limit would be 1.
    using boost::math::tools::max_value;
    return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>());
 }
 
 template <class RealType, class Policy>
 inline RealType pdf(const kolmogorov_smirnov_distribution<RealType, Policy>& dist, const RealType& x)
 {
    BOOST_FPU_EXCEPTION_GUARD
    BOOST_MATH_STD_USING  // for ADL of std functions.
 
    RealType n = dist.number_of_observations();
    RealType error_result;
    static const char* function = "boost::math::pdf(const kolmogorov_smirnov_distribution<%1%>&, %1%)";
    if(false == detail::check_x_not_NaN(function, x, &error_result, Policy()))
       return error_result;
 
    if(false == detail::check_df(function, n, &error_result, Policy()))
       return error_result;
 
    if (x < 0 || !(boost::math::isfinite)(x))
    {
       return policies::raise_domain_error<RealType>(
          function, "Kolmogorov-Smirnov parameter was %1%, but must be > 0 !", x, Policy());
    }
 
    if (2*x*x*n < constants::pi<RealType>()) {
        return detail::kolmogorov_smirnov_pdf_small_x(x, n, Policy());
    }
 
    return detail::kolmogorov_smirnov_pdf_large_x(x, n, Policy());
 } // pdf
 
 template <class RealType, class Policy>
 inline RealType cdf(const kolmogorov_smirnov_distribution<RealType, Policy>& dist, const RealType& x)
 {
     BOOST_MATH_STD_USING // for ADL of std function exp.
    static const char* function = "boost::math::cdf(const kolmogorov_smirnov_distribution<%1%>&, %1%)";
    RealType error_result;
    RealType n = dist.number_of_observations();
    if(false == detail::check_x_not_NaN(function, x, &error_result, Policy()))
       return error_result;
    if(false == detail::check_df(function, n, &error_result, Policy()))
       return error_result;
    if((x < 0) || !(boost::math::isfinite)(x)) {
       return policies::raise_domain_error<RealType>(
          function, "Random variable parameter was %1%, but must be between > 0 !", x, Policy());
    }
 
    if (x*x*n == 0)
        return 0;
 
    return jacobi_theta4tau(RealType(0), 2*x*x*n/constants::pi<RealType>(), Policy());
 } // cdf
 
 template <class RealType, class Policy>
 inline RealType cdf(const complemented2_type<kolmogorov_smirnov_distribution<RealType, Policy>, RealType>& c) {
     BOOST_MATH_STD_USING // for ADL of std function exp.
     RealType x = c.param;
    static const char* function = "boost::math::cdf(const complemented2_type<const kolmogorov_smirnov_distribution<%1%>&, %1%>)";
    RealType error_result;
    kolmogorov_smirnov_distribution<RealType, Policy> const& dist = c.dist;
    RealType n = dist.number_of_observations();
 
    if(false == detail::check_x_not_NaN(function, x, &error_result, Policy()))
       return error_result;
    if(false == detail::check_df(function, n, &error_result, Policy()))
       return error_result;
 
    if((x < 0) || !(boost::math::isfinite)(x))
       return policies::raise_domain_error<RealType>(
          function, "Random variable parameter was %1%, but must be between > 0 !", x, Policy());
 
    if (x*x*n == 0)
        return 1;
 
    if (2*x*x*n > constants::pi<RealType>())
        return -jacobi_theta4m1tau(RealType(0), 2*x*x*n/constants::pi<RealType>(), Policy());
 
    return RealType(1) - jacobi_theta4tau(RealType(0), 2*x*x*n/constants::pi<RealType>(), Policy());
 } // cdf (complemented)
 
 template <class RealType, class Policy>
 inline RealType quantile(const kolmogorov_smirnov_distribution<RealType, Policy>& dist, const RealType& p)
 {
     BOOST_MATH_STD_USING
    static const char* function = "boost::math::quantile(const kolmogorov_smirnov_distribution<%1%>&, %1%)";
    // Error check:
    RealType error_result;
    RealType n = dist.number_of_observations();
    if(false == detail::check_probability(function, p, &error_result, Policy()))
       return error_result;
    if(false == detail::check_df(function, n, &error_result, Policy()))
       return error_result;
 
    RealType k = detail::kolmogorov_smirnov_quantile_guess(p) / sqrt(n);
    const int get_digits = policies::digits<RealType, Policy>();// get digits from policy,
    std::uintmax_t m = policies::get_max_root_iterations<Policy>(); // and max iterations.
 
    return tools::newton_raphson_iterate(detail::kolmogorov_smirnov_quantile_functor<RealType, Policy>(dist, p),
            k, RealType(0), RealType(1), get_digits, m);
 } // quantile
 
 template <class RealType, class Policy>
 inline RealType quantile(const complemented2_type<kolmogorov_smirnov_distribution<RealType, Policy>, RealType>& c) {
     BOOST_MATH_STD_USING
    static const char* function = "boost::math::quantile(const kolmogorov_smirnov_distribution<%1%>&, %1%)";
    kolmogorov_smirnov_distribution<RealType, Policy> const& dist = c.dist;
    RealType n = dist.number_of_observations();
    // Error check:
    RealType error_result;
    RealType p = c.param;
 
    if(false == detail::check_probability(function, p, &error_result, Policy()))
       return error_result;
    if(false == detail::check_df(function, n, &error_result, Policy()))
       return error_result;
 
    RealType k = detail::kolmogorov_smirnov_quantile_guess(RealType(1-p)) / sqrt(n);
 
    const int get_digits = policies::digits<RealType, Policy>();// get digits from policy,
    std::uintmax_t m = policies::get_max_root_iterations<Policy>(); // and max iterations.
 
    return tools::newton_raphson_iterate(
            detail::kolmogorov_smirnov_complementary_quantile_functor<RealType, Policy>(dist, p),
            k, RealType(0), RealType(1), get_digits, m);
 } // quantile (complemented)
 
 template <class RealType, class Policy>
 inline RealType mode(const kolmogorov_smirnov_distribution<RealType, Policy>& dist)
 {
     BOOST_MATH_STD_USING
    static const char* function = "boost::math::mode(const kolmogorov_smirnov_distribution<%1%>&)";
    RealType n = dist.number_of_observations();
    RealType error_result;
    if(false == detail::check_df(function, n, &error_result, Policy()))
       return error_result;
 
     std::pair<RealType, RealType> r = boost::math::tools::brent_find_minima(
             detail::kolmogorov_smirnov_negative_pdf_functor<RealType, Policy>(),
             static_cast<RealType>(0), static_cast<RealType>(1), policies::digits<RealType, Policy>());
     return r.first / sqrt(n);
 }
 
 // Mean and variance come directly from
 // https://www.jstatsoft.org/article/view/v008i18 Section 3
 template <class RealType, class Policy>
 inline RealType mean(const kolmogorov_smirnov_distribution<RealType, Policy>& dist)
 {
     BOOST_MATH_STD_USING
    static const char* function = "boost::math::mean(const kolmogorov_smirnov_distribution<%1%>&)";
     RealType n = dist.number_of_observations();
     RealType error_result;
     if(false == detail::check_df(function, n, &error_result, Policy()))
         return error_result;
     return constants::root_half_pi<RealType>() * constants::ln_two<RealType>() / sqrt(n);
 }
 
 template <class RealType, class Policy>
 inline RealType variance(const kolmogorov_smirnov_distribution<RealType, Policy>& dist)
 {
    static const char* function = "boost::math::variance(const kolmogorov_smirnov_distribution<%1%>&)";
     RealType n = dist.number_of_observations();
     RealType error_result;
     if(false == detail::check_df(function, n, &error_result, Policy()))
         return error_result;
     return (constants::pi_sqr_div_six<RealType>()
             - constants::pi<RealType>() * constants::ln_two<RealType>() * constants::ln_two<RealType>()) / (2*n);
 }
 
 // Skewness and kurtosis come from integrating the PDF
 // The alternating series pops out a Dirichlet eta function which is related to the zeta function
 template <class RealType, class Policy>
 inline RealType skewness(const kolmogorov_smirnov_distribution<RealType, Policy>& dist)
 {
     BOOST_MATH_STD_USING
    static const char* function = "boost::math::skewness(const kolmogorov_smirnov_distribution<%1%>&)";
     RealType n = dist.number_of_observations();
     RealType error_result;
     if(false == detail::check_df(function, n, &error_result, Policy()))
         return error_result;
     RealType ex3 = RealType(0.5625) * constants::root_half_pi<RealType>() * constants::zeta_three<RealType>() / n / sqrt(n);
     RealType mean = boost::math::mean(dist);
     RealType var = boost::math::variance(dist);
     return (ex3 - 3 * mean * var - mean * mean * mean) / var / sqrt(var);
 }
 
 template <class RealType, class Policy>
 inline RealType kurtosis(const kolmogorov_smirnov_distribution<RealType, Policy>& dist)
 {
     BOOST_MATH_STD_USING
    static const char* function = "boost::math::kurtosis(const kolmogorov_smirnov_distribution<%1%>&)";
     RealType n = dist.number_of_observations();
     RealType error_result;
     if(false == detail::check_df(function, n, &error_result, Policy()))
         return error_result;
     RealType ex4 = 7 * constants::pi_sqr_div_six<RealType>() * constants::pi_sqr_div_six<RealType>() / 20 / n / n;
     RealType mean = boost::math::mean(dist);
     RealType var = boost::math::variance(dist);
     RealType skew = boost::math::skewness(dist);
     return (ex4 - 4 * mean * skew * var * sqrt(var) - 6 * mean * mean * var - mean * mean * mean * mean) / var / var;
 }
 
 template <class RealType, class Policy>
 inline RealType kurtosis_excess(const kolmogorov_smirnov_distribution<RealType, Policy>& dist)
 {
    static const char* function = "boost::math::kurtosis_excess(const kolmogorov_smirnov_distribution<%1%>&)";
     RealType n = dist.number_of_observations();
     RealType error_result;
     if(false == detail::check_df(function, n, &error_result, Policy()))
         return error_result;
     return kurtosis(dist) - 3;
 }
 }}
 #endif

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