Fast approximations

Supporting functions

• Approximations based on the normal distribution
  • Non-central chi-squared distribution: cdf and sf (Penev)
    • ctx.chi2_nc_penev_cdf()
  • (Non-central) chi-squared distribution: qtf and isf (Canal)
    • ctx.chi2_nc_canal_qtf()
  • Gamma distribution: qtf and isf (Canal)
    • ctx.gamma_canal_qtf()
  • F distribution: qtf and isf (Davis)
    • ctx.fisher_f_davis_qtf()
  • Beta distribution: qtf and isf (Davis)
    • ctx.beta_davis_qtf()
  • Pearson’s rho distribution: pdf (Winterbottom)
    • ctx.pearson_rho_wb_pdf()
  • Pearson’s rho distribution: cdf and sf (Winterbottom)
    • ctx.pearson_rho_wb_cdf()
  • Pearson’s rho distribution: qtf and isf (Winterbottom)
    • ctx.pearson_rho_wb_qtf()
  • Pearson’s rho distribution: confidence limit for \(\rho\) (Winterbottom)
    • ctx.pearson_rho_wb_cl()
  • Singly noncentral t: pdf (Broda)
    • ctx.student_t_nc_broda_pdf()
  • Singly noncentral t: cdf, sf (Broda)
    • ctx.student_t_nc_broda_cdf()
  • Singly noncentral t: qtf, isf (Harley)
    • ctx.student_t_nc_harley_qtf()
  • Singly noncentral t: confidence limit for \(\delta\) (Akahira)
    • ctx.student_t_nc_akahira_cl()
  • Doubly noncentral t: cdf, sf (Broda)
    • ctx.student_t_nc2_broda_cdf()
  • Doubly noncentral t: qtf, isf (Broda)
    • ctx.student_t_nc2_broda_qtf()
  • Spearman’s rho, first 8 cumulants (David)
    • ctx.spearman_mu8()
  • Mann-Whitney U distribution: general alternatives specified by rank order probabilities (Sundrum)
    • ctx.mannwhitney_nc_mu4()
  • First 4 moments of Kendalls \(\tau\) in the general case (Sundrum)
    • ctx.kendall_tau_nc_mu4()
• Approximations based on the chi-squared distribution
  • Non-Central chi-squared : cdf, sf (Patnaik)
    • ctx.chi2_nc_mu2_cdf()
  • Non-Central chi-squared: qtf, isf (Patnaik)
    • ctx.chi2_nc_mu2_qtf()
  • Non-Central chi-squared: confidence limit for \(\lambda\) (Winterbottom)
    • ctx.chi2_nc_wb_cl()
  • Roy’s largest root \(\theta\): pdf (Chiani)
    • ctx.roy_chiani_pdf()
  • Roy’s largest root: cdf and sf (Chiani)
    • ctx.roy_chiani_cdf()
  • Roy’s largest root: qtf and isf (Chiani)
    • ctx.roy_chiani_qtf()
• Approximations based on the central \(t\), \(F\) or beta distribution
  • Dunn-Šidák percentage points
    • ctx.dunn_sidak_qtf()
  • Singly non-central Fisher F distribution: cdf, sf (Patnaik)
    • ctx.fisher_f_nc_mu2_cdf()
  • Singly non-central F distribution: qtf, isf (Patnaik)
    • ctx.fisher_f_nc_mu2_qtf()
  • Singly non-central F: confidence interval for the noncentrality parameter \(\lambda\)
    • ctx.fisher_f_nc_mu2_cl()
  • Doubly non-central F distribution: cdf, sf (Patnaik)
    • ctx.fisher_f_nc2_mu2_cdf()
  • Doubly non-central F distribution: qtf, isf (Patnaik)
    • ctx.fisher_f_nc2_mu2_qtf()
  • Multiple correlation coefficient: cdf, sf (Lee and Gurland)
    • ctx.fisher_r2_lee_cdf()
  • Multiple correlation coefficient: qtf, isf (Lee and Gurland)
    • ctx.fisher_r2_lee_qtf()
  • Fisher \(R^2\),: confidence limit for \(\rho^2\)
    • ctx.fisher_r2_lee_cl()
  • Central Wilks’ Lambda: cdf, sf (Rao)
    • ctx.wilks_lambda_rao_cdf()
  • Central Wilks’ Lambda: qtf, isf (Rao)
    • ctx.wilks_lambda_rao_qtf()
  • Central Hotelling’s \(T^2\): cdf, sf (Pillai and Young)
    • ctx.hotelling_t2_mu3_cdf()
  • Central Hotelling’s \(T^2\): qtf, isf (Pillai and Young)
    • ctx.hotelling_t2_mu3_qtf()
  • Central Pillai’s \(V\): cdf, sf (Ginzberg)
    • ctx.pillay_v_mu3_cdf()
  • Central Pillai’s \(V\): qtf, isf (Ginzberg)
    • ctx.pillay_v_mu3_qtf()
  • Product of independent beta variables: cdf, sf (Nagarsenker)
    • ctx.beta_product_mu3_cdf()
  • Product of independent beta variables: qtf, isf (Nagarsenker)
    • ctx.beta_product_mu3_qtf()
• Approximations based on the noncentral chi-squared distribution
  • Non-central Wilks’ Lambda (GLM): cdf and sf (Fujikoshi)
    • ctx.wilks_lambda_glm_chi2_cdf()
  • Non-central Wilks’ Lambda (independence): cdf and sf (Lee)
    • ctx.wilks_lambda_ind_chi2_cdf()
  • Non-central Pillai’s V (GLM): cdf and sf Fujikoshi
    • ctx.pillai_v_glm_chi2_cdf()
  • Non-central Pillai’s V (independence): cdf and sf (Lee)
    • ctx.pillai_v_ind_chi2_cdf()
  • Non-central Hotelling \(T^2\) (GLM): cdf and sf (Fujikoshi)
    • ctx.hotelling_t2_glm_chi2_cdf()
  • Non-central Hotelling \(T^2\) (independence): cdf and sf (Lee)
    • ctx.hotelling_t2_ind_chi2_cdf()
• Approximations based on the noncentral F or beta distribution
  • Multiple correlation coefficient (Lee and Gurland)
    • ctx.fisher_r2_lee_mu3_cdf()
  • Noncentral Wilks’ Lambda under the GLM alternative
    • ctx.wilks_lambda_glm_mu2_cdf()
  • Noncentral Wilks’ Lambda under the independence alternative
    • ctx.wilks_lambda_ind_mu2_cdf()
  • Noncentral Hotelling’s T under the GLM alternative
    • ctx.hotelling_t2_glm_mu2_cdf()
  • Noncentral Hotelling’s T under the independence alternative
    • ctx.hotelling_t2_ind_mu2_cdf()
  • Noncentral Pillai’s V under the GLM alternative
    • ctx.pillai_v_glm_mu2_cdf()
  • Noncentral Pillai’s V under the independence alternative
    • ctx.pillai_v_ind_mu2_cdf()
  • Noncentral Roy’s largest root under the GLM alternative
    • ctx.roy_glm_mu2_cdf()
  • Noncentral Roy’s largest root under the independence alternative
    • ctx.roy_ind_mu2_cdf()
• Approximations based on hypergeometric functions of scalar argument
  • Hypergeometric function \({}_1F_1\) for matrix argument (Butler’s approximation)
    • ctx.hypermat_1f1_butler()
  • Comparisons with MATLAB: some results
  • Hypergeometric function \({}_2F_1\) for matrix argument (Butler’s approximation)
    • ctx.hypermat_2f1_butler()
  • Comparison with MATLAB: some results