I want to find the log-likelihood value after mle() found parameters for mixture distribution of two normal distribution

I plotted the pdf histogram of a set of data, and it looked like there are two peaks. I wanted to estimate the mixing probability, mean and standard deviation of each subpopulation, so I used a method described in this page (link below).

https://www.mathworks.com/help/stats/fitting-custom-univariate-distributions.html

Briefly, I used a method to fit custom distributions assuming the distribution is a mixture of two normal distributions. I have found the parameters that fit the distritbution reasonably well upon visual inspection. I understand that I can generate 95%CIs for these estimated parameters. However, I would also like to report a different value to evaluate the goodness of fit of this model. My understanding is that since mle() is a likelihood-based method and log-likelihood is a way to evaluate how good the model fits, I would like to find this log-likelihood value for this set of parameters. Alternatively, if anyone could suggest other ways to report the goodness of fit, it would be great.

Here is the code I used:

pdf_normmixture = @(data,p,mu1,mu2,sigma1,sigma2) ...
    p*normpdf(data,mu1,sigma1) + (1-p)*normpdf(data,mu2,sigma2);

pStart = .5;
muStart = quantile(x3,[.25 .75])

sigmaStart = sqrt(var(x3) - .25*diff(muStart).^2)

start = [pStart muStart sigmaStart sigmaStart];

lb = [0 -Inf -Inf 0 0];
ub = [1 Inf Inf Inf Inf];

options = statset('MaxIter',300,'MaxFunEvals',600);

[paramEsts,pci] = mle(data,'pdf',pdf_normmixture,'Start',start, ...
    'LowerBound',lb,'UpperBound',ub)

Thanks in advance!

Answer 1

[Ayush Modi]

Hi Alex,

As per my understanding, you would like to calculate the log-likelihood value for a set of parameters. You can achieve this using the “fitdist” function. Here is an example to demonstrate how you can accomplish this:

pd = fitdist(data,'normal');
result = pd.NLogL

Please refer to the following MathWorks documentation for more information on “fitdist” function and various distributions:

https://www.mathworks.com/help/stats/fitdist.html

Note: Select the distribution instead of “normal” as per the distribution of the “data”

I hope this resolves the issue you were facing.