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## Numerical Inversion of a Bivariate Characteristic Function

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cf2Dist2D is a MATLAB algorithm for numerical inversion of a bivariate characteristic function which evaluates PDF and CDF from bivariate CF

Updated 14 Apr 2021

cf2Dist2D calculates the CDF and PDF from the BIVARIATE characteristic function CF by using the Gil-Pelaez inversion formulae and the Riemann quadrature sum, as suggested in Shephard (1991).

The algorithm cf2Dist2D is a part of the MATLAB toolbox CharFunTool: https://github.com/witkovsky/CharFunTool

SYNTAX:
result = cf2Dist2D(cf,x)

EXAMPLE 1 (CDF/PDF of bivariate standard normal distribution)
cf = @(t) exp(-(0.9*t(:,1).^2 + 0.3*t(:,2).^2 +2*0.4*t(:,1).*t(:,2))/2);
result = cf2Dist2D(cf)
disp([result.x result.cdf])

% EXAMPLE 2 (CDF/PDF of a mixture of bivariate logistic distributions)
mu1 = [0 2];
beta1 = [1 2];
cf1 = @(t) cf2D_Logistic(t,mu1,beta1);
mu2 = [2 1];
beta2 = [2 1];
cf2 = @(t) cf2D_Logistic(t,mu2,beta2);
cf = @(t) 0.25*cf1(t) + 0.75*cf2(t);
options.xN = 51;
result = cf2Dist2D(cf,[],options)

### Cite As

Viktor Witkovsky (2021). Numerical Inversion of a Bivariate Characteristic Function (https://www.mathworks.com/matlabcentral/fileexchange/90297-numerical-inversion-of-a-bivariate-characteristic-function), MATLAB Central File Exchange. Retrieved .

##### MATLAB Release Compatibility
Created with R2020a
Compatible with any release
##### Platform Compatibility
Windows macOS Linux