To use the multivariate cumulative distribution function with Weibull, you will need to first transform your Weibull distribution to a normal distribution using the Inverse Transform Sampling method.
The Inverse Transform Sampling method uses a random variable uniformly distributed on [0, 1] to generate random variables from any distribution, in this case, the Weibull distribution. To perform this transformation, you can use the icdf (inverse cumulative distribution function) of the Weibull distribution.
Here's an updated version of your code incorporating the transformation to a normal distribution:
function Rs = parameterfun(d,X,mu_L1,sig_L1,mu_L2,sig_L2,mu_L3,sig_L3,R1,R2,R3,w1,w2,w3)
%
%Weibull parameters estimation
[k(1),lambda(1)] = wblfit(X(:,1));
[k(2),lambda(2)] = wblfit(X(:,2));
[k(3),lambda(3)] = wblfit(X(:,3));
%
%Transform Weibull to Normal using Inverse Transform Sampling
Z = norminv(wblcdf(X, lambda, k));
%
%Estimate the mean and covariance matrix for the transformed data
Z_mean = mean(Z);
Z_cov = cov(Z);
%
%Multivariate Normal cdf
p_univariate = mvncdf(Z, Z_mean, Z_cov);
%
mu_Sr1 = w1*mu_L1+wblinv(R1)*sqrt((w1*sig_L1)^2+(d(1))^2);
mu_Sr2 = w2*mu_L2+wblinv(R2)*sqrt((w2*sig_L2)^2+(d(2))^2);
mu_Sr3 = w3*mu_L3+wblinv(R3)*sqrt((w3*sig_L3)^2+(d(3))^2);
%
Y1_mean = w1*mu_L1-mu_Sr1;
Y2_mean = w2*mu_L2-mu_Sr2;
Y3_mean = w3*mu_L3-mu_Sr3;
%
Y1_std = sqrt((d(1))^2+(w1*sig_L1)^2);
Y2_std = sqrt((d(2))^2+(w2*sig_L2)^2);
Y3_std = sqrt((d(3))^2+(w3*sig_L3)^2);
%
Y_mean = [Y1_mean Y2_mean Y3_mean];
Y_std = [(Y1_std^2) (w1*w2*sig_L1*sig_L2) (w1*w3*sig_L1*sig_L3); (w2*w1*sig_L2*sig_L1) (Y2_std)^2 (w2*w3*sig_L2*sig_L3); (w3*w1*sig_L3*sig_L1) (w3*w2*sig_L3*sig_L2) (Y3_std)^2];
y = ones(size(d));
%
Rs = 1-mvncdf(y,Y_mean,Y_std).*prod(p_univariate);
%pf = 1-Rs;
%
end
In this code, I've added the transformation of the Weibull distribution to a normal distribution using the Inverse Transform Sampling method. The rest of the code remains the same. The multivariate normal CDF is then calculated using the transformed data.
Important: There may be some mistakes in this answer Experts can tell if there are any mistakes
