Fitting a stretched exponential model by Maximum likelihood

[Alexander Koplenig]

Dear Statalisters,

I would very much appreciate help regarding the following problem:


I want use ml to fit a stretched exponential function to different datasets:

y = exp(b₀ * x^{(b₁ - 1)} + b₂)


b₀, b₁ and b₂ denote the parameters of the function, while "x" and "y" are variables.

I already tried to use the nl-command to fit the function:

nl (y = exp({b0}*chars^({b1}-1)+{b2}))

In many cases, however, there are convergence issues and other problems. That's why I want to try using ml.
I have already used the ml-command, but I am not very familiar with it.

I first would like to define a program to calculate the log likelihood

program define stretched
    args lnf x
qui replace `lnf' = /* fill in likelihood function here */
end

Using this program,I would try to fit the model like this

ml model lf stretchedexp (x=y)
ml maximize

As you can see in the program above, I don't know the likelihood function of this particular stretched exponential function... unfortunately, my "statistical" skills are not sufficient in this area...

So if anyone can tell me the likelihood or has any tips, please let me know, I would be very grateful.

Many thanks

Ali

[Alexander Koplenig]

With the help of https://www.stata.com/support/faqs/s...dels-using-ml/, I managed to fill in the log-likelihood in the code above.

capture program drop mlstretched
program mlstretched
    version 14
    args lnf x b0 b1 b2 sigma
    tempvar res
    quietly gen double `res' = $ML_y1 - exp(`b0'*`x'^(`b1'-1)+`b2')
    quietly replace `lnf' = -0.5*ln(2*_pi)-ln(`sigma')-0.5*`res'^2/`sigma'^2
end

I tried to follow the FAQ description as closely as possible. However, ml does not converge, as the following test demonstrates:

/* Generate some test data */
clear
set obs 100
gen n=_n

gen rate=exp(2*n^(.6-1)+1.5)

/*Estimate the model */

ml model lf mlstretched (x: rate = n, nocons) (b1: rate = n, nocons) (b0: rate = n) (b2:) (sigma:)
ml max

I believe that there must be some error in my model specification...

If anyone has any tips, please let me know, thank you very much.

Ali