Series and integrals

Supporting functions

• Finite series algorithms for selected distributions
  • Central \(\chi^2\) distribution, cdf (integer degrees of freedom)
    • ctx.chi2_cohen_cdf()
  • Central Student \(t\) distribution, cdf (integer degrees of freedom)
    • ctx.student_t_owen_cdf()
  • Central Fisher \(F\) distribution, cdf (integer degrees of freedom)
    • ctx.fisher_f_seber_cdf()
  • Central Beta distribution, cdf (\(2a\) an integer, \(2b\) an integer)
    • ctx.beta_seber_cdf()
  • Noncentral \(\chi^2\) distribution, cdf (integer degrees of freedom)
    • ctx.chi2_nc_cohen_cdf()
  • Noncentral Student \(t\) distribution, cdf (integer degrees of freedom)
    • ctx.student_t_nc_owen_cdf()
  • Noncentral Fisher \(F\) distribution, cdf (\(m\) an even integer)
    • ctx.fisher_f_nc_seber_cdf()
  • Noncentral Beta distribution, cdf (\(b\) an integer)
    • ctx.beta_nc_seber_cdf()
  • Pearson’s \(\rho\) distribution, pdf (integer \(N\))
    • ctx.pearson_rho_nc_owen_pdf()
  • Pearson’s \(\rho\) distribution, cdf (integer \(N\))
    • ctx.pearson_rho_nc_owen_cdf()
  • Fisher’s \(R^2\) distribution, cdf (finite sum for \(N-p\) even)
    • ctx.fisher_r2_gd1_cdf()
  • Roy’s largest root distribution, pdf, cdf and sf
    • ctx.roy_pdf_cdf_sf()
• Infinite series algorithms for selected functions and distributions
  • Incomplete gamma function, continued fractions (Peizer)
    • ctx.gamma_peizer_cdf_sf_pdf()
  • Incomplete gamma function, asymptotic expansion (Paris)
    • ctx.gamma_paris_cdf_sf()
  • Incomplete beta function, continued fractions (Peizer)
    • ctx.beta_peizer_cdf_sf_pdf()
  • Noncentral \(\chi^2\) distribution, pdf, cdf and sf (Boost)
    • ctx.chi2_nc_benton_cdf_sf()
  • Noncentral Student \(t\) distribution, pdf, cdf and sf (Boost)
    • ctx.student_t_nc_benton_cdf_sf()
  • Noncentral Beta distribution, pdf, cdf and sf (Boost)
    • ctx.beta_nc_benton_cdf_sf()
  • Noncentral \(F\) distribution, pdf, cdf and sf (Benton)
    • ctx.fisher_f_nc_benton_cdf_sf()
  • Pearson’s \(\rho\) distribution, cdf and sf (Hotelling’s series)
    • ctx.pearson_rho_nc_ht_cdf()
  • Pearson’s \(\rho\) distribution, cdf and sf (Guenther’s series)
    • ctx.pearson_rho_nc_gt_cdf()
  • Fisher’s \(R^2\) distribution, cdf and sf (Boost, Benton)
    • ctx.fisher_r2_gd2_cdf()
• Finite series for lattice distributions, based on factorial moments
  • Fréchet’s formula for calculating the pmf from the factorial moments
    • ctx.frechet_pmf()
  • Laurent’s formula for calculating the cdf from the factorial moments
    • ctx.laurent_cdf()
  • Binomial distribution, pmf, cdf
    • ctx.binomial_fm_pmf()
  • Classical hypergeometric distribution, pmf, cdf
    • ctx.hypergeo_fm_pmf()
  • Wilcoxon distribution, pmf, cdf
    • ctx.wilcoxon_fm_pmf()
  • Mann-Whitney distribution, pmf, cdf
    • ctx.mannwhitney_fm_pmf()
  • Kendall distribution, pmf, cdf
    • ctx.kendall_fm_pmf()
  • Jonckheere-Terpsta \(T\) distribution, pmf, cdf
    • ctx.jterpsta_fm_pmf()
• Efficient integration of bell-shaped functions
  • quad_bellshaped()
  • Error function
    • ctx.real_quad_erf()
  • Lower non-normalised incomplete gamma function, \(\gamma(a,x)\) (Continued fractions)
    • ctx.real_quad_gamma_lower()
  • Real upper non-normalised incomplete gamma function, \(\Gamma(a,x)\) (Quadrature)
    • ctx.real_quad_gamma_upper()
  • Normalised incomplete beta function, \(I_{x}(a,b)\)
    • ctx.real_quad_ibeta()
  • Non-central chi-square cdf and sf (Chow)
    • ctx.chi_squared_nc_quad_cdf()
  • Marcum \(Q\) function
    • ctx.marcumq1()
  • Owen’s \(T\) function
    • ctx.owent()
• Verified numerical integration
  • Error function
    • ctx.real_quad_erf_verified()
  • Lower non-normalised incomplete gamma function, \(\gamma(a,x)\) (Continued fractions)
    • ctx.real_quad_gamma_lower_verified()
  • Real upper non-normalised incomplete gamma function, \(\Gamma(a,x)\) (Quadrature)
    • ctx.real_quad_gamma_upper_verified()
  • Normalised incomplete beta function, \(I_{x}(a,b)\)
    • ctx.real_quad_ibeta_verified()
  • Non-central chi-square cdf and sf (Chow)
    • ctx.chi_squared_nc_quad_cdf_verified()
  • Marcum \(Q\) function
    • ctx.marcumq1_verified()
  • Owen’s \(T\) function
    • ctx.owent_verified()