Estimating the parameters of a custom distribution using mle()

Question

I have the following code that I wish to estimate the parameters of a custom distribution. For more details on the distribution. Then using the estimated parameters I want to see if the estimated PDF resembles the distribution of the given the data (it is supposed to match the distribution of the given data).

[EDIT]: 'x' now holds a sample of data and not a PDF

The main code is:

x           = [0.0320000000000000   0.0280000000000000  0.0280000000000000  0.0270000000000000  0.0320000000000000  0.0320000000000000  0.0480000000000000  0.0890000000000000  0.0500000000000000  0.0620000000000000  0.0480000000000000  0.0300000000000000  0.0520000000000000  0.0460000000000000  0.0540000000000000  0.0520000000000000  0.0510000000000000  0.0310000000000000  0.0330000000000000  0.0330000000000000  0.0380000000000000  0.0850000000000000  0.102000000000000   0.0290000000000000  0.0530000000000000  0.0590000000000000  0.0320000000000000  0.0800000000000000  0.0410000000000000  0.0280000000000000  0.0670000000000000  0.0350000000000000  0.0420000000000000  0.0280000000000000  0.0370000000000000  0.0480000000000000  0.0330000000000000  0.101000000000000   0.0420000000000000  0.0840000000000000  0.0340000000000000  0.0900000000000000  0.0900000000000000  0.0460000000000000  0.0290000000000000  0.0330000000000000  0.0350000000000000  0.0330000000000000  0.0320000000000000  0.0420000000000000  0.0600000000000000  0.0500000000000000  0.0390000000000000  0.0480000000000000  0.0680000000000000  0.0330000000000000  0.0510000000000000  0.0430000000000000  0.0270000000000000  0.0330000000000000  0.0590000000000000  0.0380000000000000  0.0270000000000000  0.0600000000000000  0.0310000000000000  0.0520000000000000  0.0350000000000000  0.0640000000000000  0.0570000000000000  0.0520000000000000  0.0330000000000000  0.0480000000000000  0.0530000000000000  0.0380000000000000  0.0320000000000000  0.0340000000000000  0.0380000000000000  0.0470000000000000  0.0950000000000000  0.0510000000000000  0.0280000000000000  0.124000000000000   0.0360000000000000  0.0670000000000000  0.0380000000000000  0.0760000000000000  0.0440000000000000  0.0390000000000000  0.0500000000000000  0.0500000000000000  0.0370000000000000  0.0350000000000000  0.0490000000000000  0.0570000000000000  0.0560000000000000  0.0500000000000000  0.0350000000000000  0.0390000000000000  0.0390000000000000  0.0310000000000000  0.0260000000000000  0.0350000000000000  0.0610000000000000  0.0280000000000000  0.0480000000000000  0.0560000000000000  0.0650000000000000  0.0400000000000000  0.131000000000000   0.0600000000000000  0.0310000000000000  0.0620000000000000  0.0320000000000000  0.0510000000000000  0.0510000000000000  0.0480000000000000  0.0420000000000000  0.0470000000000000  0.0690000000000000  0.0590000000000000  0.0550000000000000  0.0580000000000000  0.0410000000000000  0.0320000000000000  0.0440000000000000  0.0370000000000000  0.0390000000000000  0.0620000000000000  0.0290000000000000  0.0340000000000000  0.0490000000000000  0.0300000000000000  0.0540000000000000  0.0420000000000000  0.0610000000000000  0.159000000000000   0.0520000000000000  0.0450000000000000  0.0670000000000000  0.0590000000000000  0.111000000000000   0.0250000000000000  0.0660000000000000  0.0210000000000000  0.0370000000000000  0.0330000000000000  0.0510000000000000  0.0330000000000000  0.0450000000000000  0.0530000000000000  0.0600000000000000  0.0400000000000000  0.0510000000000000  0.0680000000000000  0.0410000000000000  0.0490000000000000  0.0280000000000000  0.0520000000000000  0.0610000000000000  0.0390000000000000  0.0630000000000000  0.0360000000000000  0.0620000000000000  0.0550000000000000  0.0640000000000000  0.0650000000000000  0.0550000000000000  0.0480000000000000  0.0640000000000000  0.0850000000000000  0.0800000000000000  0.0390000000000000  0.0430000000000000  0.0380000000000000  0.0770000000000000  0.0620000000000000  0.0430000000000000  0.0490000000000000  0.0450000000000000  0.0510000000000000  0.0290000000000000  0.0440000000000000  0.0220000000000000  0.0400000000000000  0.0760000000000000  0.0290000000000000  0.0460000000000000  0.0450000000000000  0.0430000000000000  0.0790000000000000  0.0730000000000000  0.0660000000000000  0.0740000000000000  0.0330000000000000  0.0430000000000000  0.0450000000000000  0.0390000000000000  0.0610000000000000  0.0330000000000000  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0.0320000000000000  0.0590000000000000  0.0390000000000000  0.0640000000000000  0.0630000000000000  0.0310000000000000  0.0570000000000000  0.0630000000000000  0.0280000000000000  0.0560000000000000  0.0910000000000000  0.0580000000000000  0.0680000000000000  0.0420000000000000  0.0810000000000000  0.0380000000000000  0.0250000000000000  0.0340000000000000  0.0450000000000000  0.0370000000000000  0.0390000000000000  0.0300000000000000  0.0410000000000000  0.0580000000000000  0.0300000000000000  0.0830000000000000  0.0380000000000000  0.0300000000000000  0.0530000000000000  0.0610000000000000  0.0370000000000000  0.0390000000000000  0.0340000000000000  0.0280000000000000  0.0420000000000000  0.0620000000000000  0.0520000000000000  0.0310000000000000  0.0590000000000000  0.0520000000000000  0.0420000000000000  0.0430000000000000  0.0410000000000000  0.0250000000000000  0.0570000000000000  0.0370000000000000  0.0270000000000000  0.0860000000000000  0.0660000000000000  0.0470000000000000  0.0270000000000000  0.0830000000000000  0.0440000000000000  0.0680000000000000  0.0500000000000000  0.0480000000000000  0.0520000000000000  0.0510000000000000  0.0290000000000000  0.0360000000000000  0.0290000000000000  0.0390000000000000  0.0290000000000000  0.0600000000000000  0.0310000000000000  0.0440000000000000  0.0410000000000000  0.0350000000000000  0.0510000000000000  0.0660000000000000  0.0450000000000000  0.0520000000000000  0.0530000000000000  0.0260000000000000  0.0690000000000000  0.0590000000000000  0.0760000000000000  0.118000000000000   0.0500000000000000  0.0480000000000000  0.0300000000000000  0.0300000000000000  0.0390000000000000  0.0340000000000000  0.0500000000000000  0.0280000000000000  0.0410000000000000  0.0490000000000000  0.0390000000000000  0.0390000000000000  0.0420000000000000  0.0520000000000000  0.0300000000000000  0.0320000000000000  0.0650000000000000  0.0650000000000000  0.0600000000000000  0.0450000000000000  0.0680000000000000  0.0350000000000000  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0.0440000000000000  0.0390000000000000  0.0470000000000000  0.0450000000000000  0.0400000000000000  0.0880000000000000  0.0300000000000000  0.0440000000000000  0.0610000000000000  0.0370000000000000  0.0350000000000000  0.0490000000000000  0.0510000000000000  0.0820000000000000  0.0390000000000000  0.0440000000000000  0.0600000000000000  0.0310000000000000  0.0720000000000000  0.0460000000000000  0.0530000000000000  0.0500000000000000  0.0550000000000000  0.0390000000000000  0.0570000000000000  0.0460000000000000  0.0410000000000000  0.0470000000000000  0.0310000000000000  0.0720000000000000  0.0440000000000000  0.0610000000000000  0.0470000000000000  0.0420000000000000  0.0620000000000000  0.0790000000000000  0.0360000000000000  0.0450000000000000  0.0400000000000000  0.0540000000000000  0.0500000000000000  0.0430000000000000  0.0670000000000000  0.0230000000000000  0.0290000000000000  0.0450000000000000  0.0270000000000000  0.0470000000000000  0.0360000000000000  0.0600000000000000  0.0440000000000000  0.0290000000000000  0.0280000000000000  0.0350000000000000  0.0840000000000000  0.0660000000000000  0.0410000000000000  0.0300000000000000  0.0440000000000000  0.0450000000000000  0.0470000000000000  0.0620000000000000  0.0420000000000000  0.0300000000000000  0.0330000000000000  0.0320000000000000  0.0440000000000000  0.0700000000000000  0.0340000000000000  0.0420000000000000  0.0480000000000000  0.0360000000000000  0.0590000000000000  0.106000000000000   0.0280000000000000  0.0540000000000000  0.0870000000000000  0.0300000000000000  0.0300000000000000  0.0370000000000000  0.0210000000000000  0.0360000000000000  0.0910000000000000  0.126000000000000   0.0780000000000000  0.0510000000000000  0.0500000000000000  0.0370000000000000  0.0540000000000000  0.0380000000000000  0.0350000000000000  0.0480000000000000  0.0300000000000000  0.0340000000000000  0.133000000000000   0.0330000000000000  0.0340000000000000  0.0480000000000000  0.0590000000000000  0.0460000000000000  0.0650000000000000  0.0360000000000000  0.0650000000000000  0.0860000000000000  0.0290000000000000  0.0800000000000000  0.0430000000000000  0.0360000000000000  0.0490000000000000  0.0580000000000000  0.0310000000000000  0.0300000000000000  0.0330000000000000  0.0390000000000000  0.0330000000000000  0.0620000000000000  0.0330000000000000  0.0940000000000000  0.0270000000000000  0.0410000000000000  0.0570000000000000  0.0540000000000000  0.0390000000000000  0.0270000000000000  0.0590000000000000  0.0320000000000000  0.0390000000000000  0.0400000000000000  0.0720000000000000  0.0480000000000000  0.0480000000000000  0.0560000000000000  0.0730000000000000  0.0410000000000000  0.0520000000000000  0.0840000000000000  0.0590000000000000  0.0690000000000000  0.0330000000000000  0.0400000000000000  0.0320000000000000  0.0320000000000000  0.0310000000000000  0.0520000000000000  0.0760000000000000  0.0420000000000000  0.0370000000000000  0.0360000000000000  0.0780000000000000  0.0590000000000000  0.0390000000000000  0.0590000000000000  0.0880000000000000  0.0410000000000000  0.0640000000000000  0.0350000000000000  0.0350000000000000  0.0530000000000000  0.0490000000000000  0.0330000000000000  0.0640000000000000  0.0320000000000000  0.0880000000000000  0.0310000000000000  0.0980000000000000  0.0380000000000000  0.0270000000000000  0.0690000000000000  0.0530000000000000  0.0610000000000000  0.0380000000000000  0.0470000000000000  0.0620000000000000  0.0400000000000000  0.0400000000000000  0.0310000000000000  0.0740000000000000  0.0280000000000000  0.0310000000000000  0.0330000000000000  0.115000000000000   0.0890000000000000  0.0770000000000000  0.0490000000000000  0.0640000000000000  0.0580000000000000  0.0540000000000000  0.0330000000000000  0.0490000000000000  0.0230000000000000  0.0330000000000000  0.0490000000000000  0.0370000000000000  0.0330000000000000  0.0490000000000000  0.0460000000000000  0.0750000000000000  0.0420000000000000  0.0610000000000000  0.0340000000000000  0.0870000000000000  0.0390000000000000  0.0410000000000000  0.0430000000000000  0.0810000000000000  0.0570000000000000  0.0440000000000000  0.0470000000000000  0.0470000000000000  0.0320000000000000  0.0730000000000000  0.0590000000000000  0.0290000000000000  0.0450000000000000  0.0310000000000000  0.0450000000000000  0.0490000000000000  0.0880000000000000  0.0220000000000000  0.0330000000000000  0.0360000000000000  0.0350000000000000  0.0540000000000000  0.0470000000000000  0.0390000000000000  0.0300000000000000  0.0450000000000000  0.0740000000000000  0.0270000000000000  0.0320000000000000  0.0640000000000000  0.0750000000000000  0.0320000000000000  0.0300000000000000  0.0310000000000000  0.0470000000000000  0.0510000000000000  0.0520000000000000  0.0680000000000000  0.0700000000000000  0.0630000000000000  0.0410000000000000  0.0510000000000000  0.0460000000000000  0.0280000000000000  0.0780000000000000  0.0420000000000000  0.0390000000000000  0.0540000000000000  0.0310000000000000  0.0380000000000000  0.0320000000000000  0.0370000000000000  0.0260000000000000  0.0450000000000000  0.0610000000000000  0.0510000000000000  0.0240000000000000  0.0660000000000000  0.0380000000000000  0.0370000000000000  0.0450000000000000  0.0430000000000000  0.0710000000000000  0.0490000000000000  0.147000000000000   0.0340000000000000  0.0600000000000000  0.0400000000000000  0.0280000000000000  0.0350000000000000  0.0320000000000000  0.0410000000000000  0.0380000000000000  0.0380000000000000  0.0410000000000000  0.0600000000000000  0.0740000000000000  0.0350000000000000  0.0800000000000000  0.0370000000000000  0.0650000000000000  0.0300000000000000  0.0590000000000000  0.0620000000000000  0.0690000000000000  0.0540000000000000  0.0560000000000000  0.0230000000000000  0.0420000000000000  0.0560000000000000  0.0620000000000000  0.0460000000000000  0.0490000000000000  0.0690000000000000  0.0370000000000000  0.0400000000000000  0.0580000000000000  0.0340000000000000  0.0480000000000000  0.0300000000000000  0.0430000000000000  0.0270000000000000  0.0500000000000000  0.0550000000000000  0.0790000000000000  0.0380000000000000  0.0410000000000000  0.0450000000000000  0.0310000000000000  0.0430000000000000  0.0410000000000000  0.0480000000000000  0.0390000000000000  0.0650000000000000  0.0500000000000000  0.0500000000000000  0.0560000000000000  0.0690000000000000  0.0470000000000000  0.0790000000000000  0.0400000000000000  0.0540000000000000  0.0680000000000000  0.0660000000000000  0.0290000000000000  0.0520000000000000  0.0440000000000000  0.0570000000000000  0.0360000000000000  0.0610000000000000  0.0430000000000000  0.0470000000000000  0.0470000000000000  0.0500000000000000  0.0410000000000000  0.0690000000000000  0.0480000000000000  0.0480000000000000  0.0900000000000000  0.0590000000000000  0.0400000000000000  0.0340000000000000  0.0350000000000000  0.0520000000000000  0.0970000000000000  0.0350000000000000  0.0650000000000000  0.0320000000000000  0.0480000000000000  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0.0280000000000000  0.0650000000000000  0.0440000000000000  0.0210000000000000  0.0410000000000000  0.0330000000000000  0.0600000000000000  0.0810000000000000  0.0370000000000000  0.0510000000000000  0.0370000000000000  0.0940000000000000  0.0440000000000000  0.0320000000000000  0.0590000000000000  0.0400000000000000  0.0570000000000000  0.0720000000000000  0.0640000000000000  0.0610000000000000  0.0500000000000000  0.0590000000000000  0.0520000000000000  0.0510000000000000  0.0910000000000000  0.0590000000000000  0.0370000000000000  0.0320000000000000  0.0740000000000000  0.0560000000000000  0.0310000000000000  0.0390000000000000  0.112000000000000   0.0580000000000000  0.0420000000000000  0.0280000000000000  0.0340000000000000  0.0360000000000000  0.0240000000000000  0.0380000000000000  0.0640000000000000  0.0320000000000000  0.0650000000000000  0.0300000000000000  0.0270000000000000  0.0530000000000000  0.0290000000000000  0.0380000000000000  0.0360000000000000  0.0510000000000000  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0.0580000000000000  0.0540000000000000  0.0430000000000000  0.0530000000000000  0.0590000000000000  0.0520000000000000  0.0730000000000000  0.0440000000000000  0.0970000000000000  0.0520000000000000  0.0510000000000000  0.0750000000000000  0.0320000000000000  0.0480000000000000  0.0410000000000000  0.0390000000000000  0.0380000000000000  0.0430000000000000  0.0440000000000000  0.0360000000000000  0.0400000000000000  0.0480000000000000  0.0430000000000000  0.0610000000000000  0.0610000000000000  0.0370000000000000  0.0520000000000000  0.0820000000000000  0.0480000000000000  0.0330000000000000  0.0460000000000000  0.0530000000000000  0.0690000000000000  0.0370000000000000  0.0330000000000000  0.0840000000000000  0.0860000000000000  0.0470000000000000  0.102000000000000   0.0350000000000000  0.0370000000000000  0.0630000000000000  0.0760000000000000  0.0830000000000000  0.0360000000000000  0.0590000000000000  0.0430000000000000  0.0790000000000000  0.0330000000000000  0.0520000000000000  0.0530000000000000];

Censored    =  ones(1,size(x,2));% 
custpdf     =  @eval_custpdf;
custcdf     =  @eval_custcdf;
options     =  statset('Display','iter','MaxFunEvals',1000,'MaxIter',1000,...
                'FunValCheck','off','TolX',1.0e-10,'TolFun',1.0e-10);
phat        =  mle(x,'pdf', custpdf,'cdf', custcdf,...
               'start',[0.6,0.02,1.01,2,4,-10],...
               'lowerbound',[0 0 0 0 0 -inf],...
               'upperbound',[inf inf inf inf inf inf],...
               'Censoring',Censored,...
               'Options',options);;

% Checking how close the estimated PDF and CDF match with those from the data x
figure();
h        =  histogram(x,'Normalization','probability');hold on
x_times  =  h.BinEdges(1:end-1) + h.BinWidth/2 ;
y_vals   =  custpdf(x_times,  phat(1), phat(2), phat(3), phat(4), phat(5), phat(6))./...
           sum(custpdf(x_times,  phat(1), phat(2), phat(3), phat(4), phat(5), phat(6)),'omitnan');
plot(x_times,y_vals,'linewidth',2)
legend('Data','Estimated PDF')

The functions are:

function out = eval_custpdf(x,myalpha,mytheta,mybeta,a,b,c)
  first_integral      =  integral(@(x) eval_K(x,a,b,c),0,1).^-1;
  theta_t_ratio       =  (mytheta./x);
  incomplete_gamma    =  igamma(myalpha,theta_t_ratio.^mybeta);
  n_gamma             =  gamma(myalpha);
  exponent_term       =  exp(-theta_t_ratio.^mybeta-(c.*(incomplete_gamma./n_gamma)));


  numerator   =  first_integral.* mybeta.*incomplete_gamma.^(a-1).*...
               theta_t_ratio.^(myalpha*mybeta+1).*exponent_term;
  denominator =  mytheta.* n_gamma.^(a+b-1).* (n_gamma-incomplete_gamma.^mybeta).^(1-b);

  out         =  numerator./denominator;
end

function out = eval_custcdf(x,myalpha,mytheta,mybeta,a,b,c)
 out = zeros(size(x));
   for i = 1: length(x)
     first_integral      =  integral(@(x) eval_K(x,a,b,c),0,1).^-1;
     theta_t_ratio       =  mytheta./x(i);
     incomplete_gamma    =  igamma(myalpha,theta_t_ratio.^mybeta);
     n_gamma             =  gamma(myalpha);
     second_integral     =  integral(@(x) eval_K(x,a,b,c),0,...
                            incomplete_gamma.^mybeta./n_gamma);
    %  second_integral     =  integral(@(x) eval_K(x,a,b,c),0,2);
     out(i)     =  first_integral*second_integral;
   end
end 

function out = eval_K(x,a,b,c)

  out =  x.^(a-1).*(1-x).^(b-1).*exp(-c.*x);

end

However, I have not been successfull to obtain the desired PDF. As you can see in the figure, the estimated PDF (orange line) does not trace the histogram of 'x' (blue bars).

[UPDATE] Normalized histogram

Note that I have varied the initial values of the parameters. But this is very time consuming. I also increased the number of iterations and minimized the tolerance, but no luck yet. Is there a better way to estimate the paratemters other than mle?

Any help would be appreciated.

Thanks in advance.

Answer 1

Noting this from the help

If the 'censoring' name/value pair is not present, you may omit the 'cdf' name/value pair.

gives us a first suggestion for debugging. So removing the censoring part and the CDF from the input list and running

phat =  mle(x,'pdf', @eval_custpdf,'start',[0.6,0.02,1.01,2,4,-10]);
phat =  mle(x,'pdf', @eval_custpdf,'start',phat); %Restart for better result

yields the figure

Telling us that the problem might be in the CDF function. Comparing with the link given in the question, we see that this line

second_integral     =  integral(@(x) eval_K(x,a,b,c),0,incomplete_gamma.^mybeta./n_gamma);

should have been

second_integral =  integral(@(x) eval_K(x,a,b,c),0,incomplete_gamma./n_gamma);