| // wald_example.cpp or inverse_gaussian_example.cpp |
| |
| // Copyright Paul A. Bristow 2010. |
| |
| // 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) |
| |
| // Example of using the Inverse Gaussian (or Inverse Normal) distribution. |
| // The Wald Distribution is |
| |
| |
| // Note that this file contains Quickbook mark-up as well as code |
| // and comments, don't change any of the special comment mark-ups! |
| |
| //[inverse_gaussian_basic1 |
| /*` |
| First we need some includes to access the normal distribution |
| (and some std output of course). |
| */ |
| |
| #ifdef _MSC_VER |
| # pragma warning (disable : 4224) |
| # pragma warning (disable : 4189) |
| # pragma warning (disable : 4100) |
| # pragma warning (disable : 4224) |
| # pragma warning (disable : 4512) |
| # pragma warning (disable : 4702) |
| # pragma warning (disable : 4127) |
| #endif |
| |
| //#define BOOST_MATH_INSTRUMENT |
| |
| #define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error |
| #define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error |
| |
| #include <boost/math/distributions/inverse_gaussian.hpp> // for inverse_gaussian_distribution |
| using boost::math::inverse_gaussian; // typedef provides default type is double. |
| using boost::math::inverse_gaussian_distribution; // for inverse gaussian distribution. |
| |
| #include <boost/math/distributions/normal.hpp> // for normal_distribution |
| using boost::math::normal; // typedef provides default type is double. |
| |
| #include <boost/array.hpp> |
| using boost::array; |
| |
| #include <iostream> |
| using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint; |
| #include <iomanip> |
| using std::setw; using std::setprecision; |
| #include <limits> |
| using std::numeric_limits; |
| #include <sstream> |
| using std::string; |
| #include <string> |
| using std::stringstream; |
| |
| // const double tol = 3 * numeric_limits<double>::epsilon(); |
| |
| int main() |
| { |
| cout << "Example: Inverse Gaussian Distribution."<< endl; |
| |
| try |
| { |
| |
| double tolfeweps = numeric_limits<double>::epsilon(); |
| //cout << "Tolerance = " << tol << endl; |
| |
| int precision = 17; // traditional tables are only computed to much lower precision. |
| cout.precision(17); // std::numeric_limits<double>::max_digits10; for 64-bit doubles. |
| |
| // Traditional tables and values. |
| double step = 0.2; // in z |
| double range = 4; // min and max z = -range to +range. |
| // Construct a (standard) inverse gaussian distribution s |
| inverse_gaussian w11(1, 1); |
| // (default mean = units, and standard deviation = unity) |
| cout << "(Standard) Inverse Gaussian distribution, mean = "<< w11.mean() |
| << ", scale = " << w11.scale() << endl; |
| |
| /*` First the probability distribution function (pdf). |
| */ |
| cout << "Probability distribution function (pdf) values" << endl; |
| cout << " z " " pdf " << endl; |
| cout.precision(5); |
| for (double z = (numeric_limits<double>::min)(); z < range + step; z += step) |
| { |
| cout << left << setprecision(3) << setw(6) << z << " " |
| << setprecision(precision) << setw(12) << pdf(w11, z) << endl; |
| } |
| cout.precision(6); // default |
| /*`And the area under the normal curve from -[infin] up to z, |
| the cumulative distribution function (cdf). |
| */ |
| |
| // For a (default) inverse gaussian distribution. |
| cout << "Integral (area under the curve) from 0 up to z (cdf) " << endl; |
| cout << " z " " cdf " << endl; |
| for (double z = (numeric_limits<double>::min)(); z < range + step; z += step) |
| { |
| cout << left << setprecision(3) << setw(6) << z << " " |
| << setprecision(precision) << setw(12) << cdf(w11, z) << endl; |
| } |
| /*`giving the following table: |
| [pre |
| z pdf |
| 2.23e-308 -1.#IND |
| 0.2 0.90052111680384117 |
| 0.4 1.0055127039453111 |
| 0.6 0.75123750098955733 |
| 0.8 0.54377310461643302 |
| 1 0.3989422804014327 |
| 1.2 0.29846949816803292 |
| 1.4 0.2274579835638664 |
| 1.6 0.17614566625628389 |
| 1.8 0.13829083543591469 |
| 2 0.10984782236693062 |
| 2.2 0.088133964251182237 |
| 2.4 0.071327382959107177 |
| 2.6 0.058162562161661699 |
| 2.8 0.047742223328567722 |
| 3 0.039418357969819712 |
| 3.2 0.032715223861241892 |
| 3.4 0.027278388940958308 |
| 3.6 0.022840312999395804 |
| 3.8 0.019196657941016954 |
| 4 0.016189699458236451 |
| Integral (area under the curve) from 0 up to z (cdf) |
| z cdf |
| 2.23e-308 0 |
| 0.2 0.063753567519976254 |
| 0.4 0.2706136704424541 |
| 0.6 0.44638391340412931 |
| 0.8 0.57472390962590925 |
| 1 0.66810200122317065 |
| 1.2 0.73724578422952536 |
| 1.4 0.78944214237790356 |
| 1.6 0.82953458108474554 |
| 1.8 0.86079282968276671 |
| 2 0.88547542598600626 |
| 2.2 0.90517870624273966 |
| 2.4 0.92105495653509362 |
| 2.6 0.93395164268166786 |
| 2.8 0.94450240360053817 |
| 3 0.95318792074278835 |
| 3.2 0.96037753019309191 |
| 3.4 0.96635823989417369 |
| 3.6 0.97135533107998406 |
| 3.8 0.97554722413538364 |
| 4 0.97907636417888622 |
| ] |
| |
| /*`We can get the inverse, the quantile, percentile, percentage point, or critical value |
| for a probability for a few probability from the above table, for z = 0.4, 1.0, 2.0: |
| */ |
| cout << quantile(w11, 0.27061367044245421 ) << endl; // 0.4 |
| cout << quantile(w11, 0.66810200122317065) << endl; // 1.0 |
| cout << quantile(w11, 0.88547542598600615) << endl; // 2.0 |
| /*`turning the expect values apart from some 'computational noise' in the least significant bit or two. |
| |
| [pre |
| 0.40000000000000008 |
| 0.99999999999999967 |
| 1.9999999999999973 |
| ] |
| |
| */ |
| |
| // cout << "pnorm01(-0.406053) " << pnorm01(-0.406053) << ", cdfn01(-0.406053) = " << cdf(n01, -0.406053) << endl; |
| //cout << "pnorm01(0.5) = " << pnorm01(0.5) << endl; // R pnorm(0.5,0,1) = 0.6914625 == 0.69146246127401312 |
| // R qnorm(0.6914625,0,1) = 0.5 |
| |
| // formatC(SuppDists::qinvGauss(0.3649755481729598, 1, 1), digits=17) [1] "0.50000000969034875" |
| |
| |
| |
| // formatC(SuppDists::dinvGauss(0.01, 1, 1), digits=17) [1] "2.0811768202028392e-19" |
| // formatC(SuppDists::pinvGauss(0.01, 1, 1), digits=17) [1] "4.122313403318778e-23" |
| |
| |
| |
| //cout << " qinvgauss(0.3649755481729598, 1, 1) = " << qinvgauss(0.3649755481729598, 1, 1) << endl; // 0.5 |
| // cout << quantile(s, 0.66810200122317065) << endl; // expect 1, get 0.50517388467190727 |
| //cout << " qinvgauss(0.62502320258649202, 1, 1) = " << qinvgauss(0.62502320258649202, 1, 1) << endl; // 0.9 |
| //cout << " qinvgauss(0.063753567519976254, 1, 1) = " << qinvgauss(0.063753567519976254, 1, 1) << endl; // 0.2 |
| //cout << " qinvgauss(0.0040761113207110162, 1, 1) = " << qinvgauss(0.0040761113207110162, 1, 1) << endl; // 0.1 |
| |
| //double x = 1.; // SuppDists::pinvGauss(0.4, 1,1) [1] 0.2706137 |
| //double c = pinvgauss(x, 1, 1); // 0.3649755481729598 == cdf(x, 1,1) 0.36497554817295974 |
| //cout << " pinvgauss(x, 1, 1) = " << c << endl; // pinvgauss(x, 1, 1) = 0.27061367044245421 |
| //double p = pdf(w11, x); |
| //double c = cdf(w11, x); // cdf(1, 1, 1) = 0.66810200122317065 |
| //cout << "cdf(" << x << ", " << w11.mean() << ", "<< w11.scale() << ") = " << c << endl; // cdf(x, 1, 1) 0.27061367044245421 |
| //cout << "pdf(" << x << ", " << w11.mean() << ", "<< w11.scale() << ") = " << p << endl; |
| //double q = quantile(w11, c); |
| //cout << "quantile(w11, " << c << ") = " << q << endl; |
| |
| //cout << "quantile(w11, 4.122313403318778e-23) = "<< quantile(w11, 4.122313403318778e-23) << endl; // quantile |
| //cout << "quantile(w11, 4.8791443010851493e-219) = " << quantile(w11, 4.8791443010851493e-219) << endl; // quantile |
| |
| //double c1 = 1 - cdf(w11, x); // 1 - cdf(1, 1, 1) = 0.33189799877682935 |
| //cout << "1 - cdf(" << x << ", " << w11.mean() << ", " << w11.scale() << ") = " << c1 << endl; // cdf(x, 1, 1) 0.27061367044245421 |
| //double cc = cdf(complement(w11, x)); |
| //cout << "cdf(complement(" << x << ", " << w11.mean() << ", "<< w11.scale() << ")) = " << cc << endl; // cdf(x, 1, 1) 0.27061367044245421 |
| //// 1 - cdf(1000, 1, 1) = 0 |
| //// cdf(complement(1000, 1, 1)) = 4.8694344366900402e-222 |
| |
| //cout << "quantile(w11, " << c << ") = "<< quantile(w11, c) << endl; // quantile = 0.99999999999999978 == x = 1 |
| //cout << "quantile(w11, " << c << ") = "<< quantile(w11, 1 - c) << endl; // quantile complement. quantile(w11, 0.66810200122317065) = 0.46336593652340152 |
| // cout << "quantile(complement(w11, " << c << ")) = " << quantile(complement(w11, c)) << endl; // quantile complement = 0.46336593652340163 |
| |
| // cdf(1, 1, 1) = 0.66810200122317065 |
| // 1 - cdf(1, 1, 1) = 0.33189799877682935 |
| // cdf(complement(1, 1, 1)) = 0.33189799877682929 |
| |
| // quantile(w11, 0.66810200122317065) = 0.99999999999999978 |
| // 1 - quantile(w11, 0.66810200122317065) = 2.2204460492503131e-016 |
| // quantile(complement(w11, 0.33189799877682929)) = 0.99999999999999989 |
| |
| |
| // qinvgauss(c, 1, 1) = 0.3999999999999998 |
| // SuppDists::qinvGauss(0.270613670442454, 1, 1) [1] 0.4 |
| |
| |
| /* |
| double qs = pinvgaussU(c, 1, 1); // |
| cout << "qinvgaussU(c, 1, 1) = " << qs << endl; // qinvgaussU(c, 1, 1) = 0.86567442459240929 |
| // > z=q - exp(c) * p [1] 0.8656744 qs 0.86567442459240929 double |
| // Is this the complement? |
| cout << "qgamma(0.2, 0.5, 1) expect 0.0320923 = " << qgamma(0.2, 0.5, 1) << endl; |
| // qgamma(0.2, 0.5, 1) expect 0.0320923 = 0.032092377333650807 |
| |
| |
| cout << "qinvgauss(pinvgauss(x, 1, 1) = " << q |
| << ", diff = " << x - q << ", fraction = " << (x - q) /x << endl; // 0.5 |
| |
| */ // > SuppDists::pinvGauss(0.02, 1,1) [1] 4.139176e-12 |
| // > SuppDists::qinvGauss(4.139176e-12, 1,1) [1] 0.02000000 |
| |
| |
| // pinvGauss(1,1,1) = 0.668102 C++ == 0.66810200122317065 |
| // qinvGauss(0.668102,1,1) = 1 |
| |
| // SuppDists::pinvGauss(0.3,1,1) = 0.1657266 |
| // cout << "qinvGauss(0.0040761113207110162, 1, 1) = " << qinvgauss(0.0040761113207110162, 1, 1) << endl; |
| //cout << "quantile(s, 0.1657266) = " << quantile(s, 0.1657266) << endl; // expect 1. |
| |
| //wald s12(2, 1); |
| //cout << "qinvGauss(0.3, 2, 1) = " << qinvgauss(0.3, 2, 1) << endl; // SuppDists::qinvGauss(0.3,2,1) == 0.58288065635052944 |
| //// but actually get qinvGauss(0.3, 2, 1) = 0.58288064777632187 |
| //cout << "cdf(s12, 0.3) = " << cdf(s12, 0.3) << endl; // cdf(s12, 0.3) = 0.10895339868447573 |
| |
| // using boost::math::wald; |
| //cout.precision(6); |
| |
| /* |
| double m = 1; |
| double l = 1; |
| double x = 0.1; |
| //c = cdf(w, x); |
| double p = pinvgauss(x, m, l); |
| cout << "x = " << x << ", pinvgauss(x, m, l) = " << p << endl; // R 0.4 0.2706137 |
| double qg = qgamma(1.- p, 0.5, 1.0, true, false); |
| cout << " qgamma(1.- p, 0.5, 1.0, true, false) = " << qg << endl; // 0.606817 |
| double g = guess_whitmore(p, m, l); |
| cout << "m = " << m << ", l = " << l << ", x = " << x << ", guess = " << g |
| << ", diff = " << (x - g) << endl; |
| |
| g = guess_wheeler(p, m, l); |
| cout << "m = " << m << ", l = " << l << ", x = " << x << ", guess = " << g |
| << ", diff = " << (x - g) << endl; |
| |
| g = guess_bagshaw(p, m, l); |
| cout << "m = " << m << ", l = " << l << ", x = " << x << ", guess = " << g |
| << ", diff = " << (x - g) << endl; |
| |
| // m = 1, l = 10, x = 0.9, guess = 0.89792, diff = 0.00231075 so a better fit. |
| // x = 0.9, guess = 0.887907 |
| // x = 0.5, guess = 0.474977 |
| // x = 0.4, guess = 0.369597 |
| // x = 0.2, guess = 0.155196 |
| |
| // m = 1, l = 2, x = 0.9, guess = 1.0312, diff = -0.145778 |
| // m = 1, l = 2, x = 0.1, guess = 0.122201, diff = -0.222013 |
| // m = 1, l = 2, x = 0.2, guess = 0.299326, diff = -0.49663 |
| // m = 1, l = 2, x = 0.5, guess = 1.00437, diff = -1.00875 |
| // m = 1, l = 2, x = 0.7, guess = 1.01517, diff = -0.450247 |
| |
| double ls[7] = {0.1, 0.2, 0.5, 1., 2., 10, 100}; // scale values. |
| double ms[10] = {0.001, 0.02, 0.1, 0.2, 0.5, 0.9, 1., 2., 10, 100}; // mean values. |
| */ |
| |
| cout.precision(6); // Restore to default. |
| } // try |
| catch(const std::exception& e) |
| { // Always useful to include try & catch blocks because default policies |
| // are to throw exceptions on arguments that cause errors like underflow, overflow. |
| // Lacking try & catch blocks, the program will abort without a message below, |
| // which may give some helpful clues as to the cause of the exception. |
| std::cout << |
| "\n""Message from thrown exception was:\n " << e.what() << std::endl; |
| } |
| return 0; |
| } // int main() |
| |
| |
| /* |
| |
| Output is: |
| |
| inverse_gaussian_example.cpp |
| inverse_gaussian_example.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Debug\inverse_gaussian_example.exe |
| Example: Inverse Gaussian Distribution. |
| (Standard) Inverse Gaussian distribution, mean = 1, scale = 1 |
| Probability distribution function (pdf) values |
| z pdf |
| 2.23e-308 -1.#IND |
| 0.2 0.90052111680384117 |
| 0.4 1.0055127039453111 |
| 0.6 0.75123750098955733 |
| 0.8 0.54377310461643302 |
| 1 0.3989422804014327 |
| 1.2 0.29846949816803292 |
| 1.4 0.2274579835638664 |
| 1.6 0.17614566625628389 |
| 1.8 0.13829083543591469 |
| 2 0.10984782236693062 |
| 2.2 0.088133964251182237 |
| 2.4 0.071327382959107177 |
| 2.6 0.058162562161661699 |
| 2.8 0.047742223328567722 |
| 3 0.039418357969819712 |
| 3.2 0.032715223861241892 |
| 3.4 0.027278388940958308 |
| 3.6 0.022840312999395804 |
| 3.8 0.019196657941016954 |
| 4 0.016189699458236451 |
| Integral (area under the curve) from 0 up to z (cdf) |
| z cdf |
| 2.23e-308 0 |
| 0.2 0.063753567519976254 |
| 0.4 0.2706136704424541 |
| 0.6 0.44638391340412931 |
| 0.8 0.57472390962590925 |
| 1 0.66810200122317065 |
| 1.2 0.73724578422952536 |
| 1.4 0.78944214237790356 |
| 1.6 0.82953458108474554 |
| 1.8 0.86079282968276671 |
| 2 0.88547542598600626 |
| 2.2 0.90517870624273966 |
| 2.4 0.92105495653509362 |
| 2.6 0.93395164268166786 |
| 2.8 0.94450240360053817 |
| 3 0.95318792074278835 |
| 3.2 0.96037753019309191 |
| 3.4 0.96635823989417369 |
| 3.6 0.97135533107998406 |
| 3.8 0.97554722413538364 |
| 4 0.97907636417888622 |
| 0.40000000000000008 |
| 0.99999999999999967 |
| 1.9999999999999973 |
| |
| |
| |
| > SuppDists::dinvGauss(2, 1, 1) [1] 0.1098478 |
| > SuppDists::dinvGauss(0.4, 1, 1) [1] 1.005513 |
| > SuppDists::dinvGauss(0.5, 1, 1) [1] 0.8787826 |
| > SuppDists::dinvGauss(0.39, 1, 1) [1] 1.016559 |
| > SuppDists::dinvGauss(0.38, 1, 1) [1] 1.027006 |
| > SuppDists::dinvGauss(0.37, 1, 1) [1] 1.036748 |
| > SuppDists::dinvGauss(0.36, 1, 1) [1] 1.045661 |
| > SuppDists::dinvGauss(0.35, 1, 1) [1] 1.053611 |
| > SuppDists::dinvGauss(0.3, 1, 1) [1] 1.072888 |
| > SuppDists::dinvGauss(0.1, 1, 1) [1] 0.2197948 |
| > SuppDists::dinvGauss(0.2, 1, 1) [1] 0.9005211 |
| > |
| x = 0.3 [1, 1] 1.0728879234594337 // R SuppDists::dinvGauss(0.3, 1, 1) [1] 1.072888 |
| |
| x = 1 [1, 1] 0.3989422804014327 |
| |
| |
| 0 " NA" |
| 1 "0.3989422804014327" |
| 2 "0.10984782236693059" |
| 3 "0.039418357969819733" |
| 4 "0.016189699458236468" |
| 5 "0.007204168934430732" |
| 6 "0.003379893528659049" |
| 7 "0.0016462878258114036" |
| 8 "0.00082460931140859956" |
| 9 "0.00042207355643694234" |
| 10 "0.00021979480031862676" |
| |
| |
| [1] " NA" " 0.690988298942671" "0.11539974210409144" |
| [4] "0.01799698883772935" "0.0029555399206496469" "0.00050863023587406587" |
| [7] "9.0761842931362914e-05" "1.6655279133132795e-05" "3.1243174913715429e-06" |
| [10] "5.96530227727434e-07" "1.1555606328299836e-07" |
| |
| |
| matC(dinvGauss(0:10, 1, 3), digits=17) df = 3 |
| [1] " NA" " 0.690988298942671" "0.11539974210409144" |
| [4] "0.01799698883772935" "0.0029555399206496469" "0.00050863023587406587" |
| [7] "9.0761842931362914e-05" "1.6655279133132795e-05" "3.1243174913715429e-06" |
| [10] "5.96530227727434e-07" "1.1555606328299836e-07" |
| $title |
| [1] "Inverse Gaussian" |
| |
| $nu |
| [1] 1 |
| |
| $lambda |
| [1] 3 |
| |
| $Mean |
| [1] 1 |
| |
| $Median |
| [1] 0.8596309 |
| |
| $Mode |
| [1] 0.618034 |
| |
| $Variance |
| [1] 0.3333333 |
| |
| $SD |
| [1] 0.5773503 |
| |
| $ThirdCentralMoment |
| [1] 0.3333333 |
| |
| $FourthCentralMoment |
| [1] 0.8888889 |
| |
| $PearsonsSkewness...mean.minus.mode.div.SD |
| [1] 0.6615845 |
| |
| $Skewness...sqrtB1 |
| [1] 1.732051 |
| |
| $Kurtosis...B2.minus.3 |
| [1] 5 |
| |
| Example: Wald distribution. |
| (Standard) Wald distribution, mean = 1, scale = 1 |
| 1 dx = 0.24890250442652451, x = 0.70924622051646713 |
| 2 dx = -0.038547954953794553, x = 0.46034371608994262 |
| 3 dx = -0.0011074090830291131, x = 0.49889167104373716 |
| 4 dx = -9.1987259926368029e-007, x = 0.49999908012676625 |
| 5 dx = -6.346513344581067e-013, x = 0.49999999999936551 |
| dx = 6.3168242705156857e-017 at i = 6 |
| qinvgauss(0.3649755481729598, 1, 1) = 0.50000000000000011 |
| 1 dx = 0.6719944578376621, x = 1.3735318786222666 |
| 2 dx = -0.16997432635769361, x = 0.70153742078460446 |
| 3 dx = -0.027865119206495724, x = 0.87151174714229807 |
| 4 dx = -0.00062283290009492603, x = 0.89937686634879377 |
| 5 dx = -3.0075104108208687e-007, x = 0.89999969924888867 |
| 6 dx = -7.0485322513588089e-014, x = 0.89999999999992975 |
| 7 dx = 9.557331866250277e-016, x = 0.90000000000000024 |
| dx = 0 at i = 8 |
| qinvgauss(0.62502320258649202, 1, 1) = 0.89999999999999925 |
| 1 dx = -0.0052296256747447678, x = 0.19483508278446249 |
| 2 dx = 6.4699046853900505e-005, x = 0.20006470845920726 |
| 3 dx = 9.4123530465288137e-009, x = 0.20000000941235335 |
| 4 dx = 2.7739513919147025e-016, x = 0.20000000000000032 |
| dx = 1.5410841066192808e-016 at i = 5 |
| qinvgauss(0.063753567519976254, 1, 1) = 0.20000000000000004 |
| 1 dx = -1, x = -0.46073286697416105 |
| 2 dx = 0.47665501251497061, x = 0.53926713302583895 |
| 3 dx = -0.171105768635964, x = 0.062612120510868341 |
| 4 dx = 0.091490360797512563, x = 0.23371788914683234 |
| 5 dx = 0.029410311722649803, x = 0.14222752834931979 |
| 6 dx = 0.010761845493592421, x = 0.11281721662666999 |
| 7 dx = 0.0019864890597643035, x = 0.10205537113307757 |
| 8 dx = 6.8800383732599561e-005, x = 0.10006888207331327 |
| 9 dx = 8.1689466405590418e-008, x = 0.10000008168958067 |
| 10 dx = 1.133634672475146e-013, x = 0.10000000000011428 |
| 11 dx = 5.9588135045224526e-016, x = 0.10000000000000091 |
| 12 dx = 3.433223674791152e-016, x = 0.10000000000000031 |
| dx = 9.0763384505974048e-017 at i = 13 |
| qinvgauss(0.0040761113207110162, 1, 1) = 0.099999999999999964 |
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| wald_example.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Debug\wald_example.exe |
| Example: Wald distribution. |
| Tolerance = 6.66134e-016 |
| (Standard) Wald distribution, mean = 1, scale = 1 |
| cdf(x, 1,1) 4.1390252102096375e-012 |
| qinvgauss(pinvgauss(x, 1, 1) = 0.020116801973767886, diff = -0.00011680197376788548, fraction = -0.005840098688394274 |
| ____________________________________________________________ |
| wald 1, 1 |
| x = 0.02, diff x - qinvgauss(cdf) = -0.00011680197376788548 |
| x = 0.10000000000000001, diff x - qinvgauss(cdf) = 8.7430063189231078e-016 |
| x = 0.20000000000000001, diff x - qinvgauss(cdf) = -1.1102230246251565e-016 |
| x = 0.29999999999999999, diff x - qinvgauss(cdf) = 0 |
| x = 0.40000000000000002, diff x - qinvgauss(cdf) = 2.2204460492503131e-016 |
| x = 0.5, diff x - qinvgauss(cdf) = -1.1102230246251565e-016 |
| x = 0.59999999999999998, diff x - qinvgauss(cdf) = 1.1102230246251565e-016 |
| x = 0.80000000000000004, diff x - qinvgauss(cdf) = 1.1102230246251565e-016 |
| x = 0.90000000000000002, diff x - qinvgauss(cdf) = 0 |
| x = 0.98999999999999999, diff x - qinvgauss(cdf) = -1.1102230246251565e-016 |
| x = 0.999, diff x - qinvgauss(cdf) = -1.1102230246251565e-016 |
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| */ |
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