matlab - Using MLE function to estimate the parameters of a custom distribution

Question

I am trying to use mle() function in MATLAB to estimate the parameters of a 6-parameter custom distribution.

The PDF of the custom distribution is

and the CDF is

where Γ(x,y) and Γ(x) are the upper incomplete gamma function and the gamma function, respectively. α, θ, β, a, b, and c are the parameters of the custom distribution. K is given by

Given a data vector 'data', I want to estimate the parameters α, θ, β, a, b, and c.

So, far I have come up with this code:

data        =  rand(20000,1); % Since I cannot upload the acutal data, we may use this
t           =  0:0.0001:0.5;    
fun         =  @(w,a,b,c) w^(a-1)*(1-w)^(b-1)*exp^(-c*w);

% to estimate the parameters
custpdf     =  @(data,myalpha,mybeta,mytheta,a,b,c)...
                ((integral(@(t)fun(t,a,b,c),0,1)^-1)*...
                mybeta*...
                igamma(myalpha,((mytheta/t)^mybeta)^(a-1))*...
                (mytheta/t)^(myalpha*mybeta+1)*...
                exp(-(mytheta/t)^mybeta-(c*(igamma(myalpha,(mytheta/t)^mybeta)/gamma(myalpha)))))...
                /...
                (mytheta*...
                gamma(myalpha)^(a+b-1)*...
                (gamma(myalpha)-igamma(myalpha,(mytheta/t)^mybeta))^(1-b));

custcdf     =  @(data,myalpha,mybeta,mytheta,a,b,c)...
                (integral(@(t)fun(t,a,b,c),0,1)^-1)*...
                integral(@(t)fun(t,a,b,c),0,igamma(myalpha,(mytheta/t)^mybeta)^mybeta/gamma(myalpha));

phat        =  mle(data,'pdf',custpdf,'cdf',custcdf,'start',0.0);

But I get the following error:

Error using mlecustom (line 166)
Error evaluating the user-supplied pdf function
'@(data,myalpha,mybeta,mytheta,a,b,c)((integral(@(t)fun(t,a,b,c),0,1)^-1)*mybeta*igamma(myalpha,((mytheta/t)^mybeta)^(a-1))*(mytheta/t)^(myalpha*mybeta+1)*exp(-(mytheta/t)^mybeta-(c*(igamma(myalpha,(mytheta/t)^mybeta)/gamma(myalpha)))))/(mytheta*gamma(myalpha)^(a+b-1)*(gamma(myalpha)-igamma(myalpha,(mytheta/t)^mybeta))^(1-b))'.

Error in mle (line 245)
            phat = mlecustom(data,varargin{:});

Caused by:
    Not enough input arguments.

I tried to look into the error lines but I can't figure out where the error actually is.

Which function lacks fewer inputs? Is it referring to fun? Why would mle lack fewer inputs when it is trying to estimate the parameters?

Could someone kindly help me debug the error?

Thanks in advance.

Answer 1

• exp() is a function, not a variable, precise the argument

exp^(-c*w) ---> exp(-c*w)

• The starting point concerns the 6 parameters, not only one 0.1*ones(1,6)
• In custcdf mle requires the upper bound of the integral to be a scalar, I did some trial and error and the range is [2~9]. for the trial some values lead to negative cdf or less than 1 discard them.
• Then use the right one to compute the upper bound see if it's the same as the one you predefined.

I re-write all the functions, check them out

The code is as follow

Censored = ones(5,1);% All data could be trusted 

data        =  rand(5,1); % Since I cannot upload the acutal data, we may use this

f         =  @(w,a,b,c) (w.^(a-1)).*((1-w).^(b-1)).*exp(-c.*w);
% to estimate the parameters
custpdf     =  @(t,alpha,theta,beta, a,b,c)...
                (((integral(@(w)f(w,a,b,c), 0,1)).^-1).*...
                beta.*...
                ((igamma(alpha, (theta./t).^beta)).^(a-1)).*...
                ((theta./t).^(alpha.*beta + 1 )).*...
                exp(-(((theta./t).^beta)+...
                c.*igamma(alpha, (theta./t).^beta)./gamma(alpha))))./...
                (theta.*...
                ((gamma(alpha)).^(a+b-1)).*...
                 ((gamma(alpha)-...
                 igamma(alpha, (theta./t).^beta)).^(1-b)));


custcdf = @(t,alpha,theta,beta, a,b,c)...
         ((integral(@(w)f(w,a,b,c), 0,1)).^-1).*...         
     (integral(@(w)f(w,a,b,c), 0,2));



phat = mle(data,'pdf',custpdf,'cdf',custcdf,'start', 0.1.*ones(1,6),'Censoring',Censored);

Result

    phat = 0.1017    0.1223    0.1153    0.1493   -0.0377    0.0902