Dear Dr Porotsky

I have a code on R to find the goodness of fit through maximum likelihood estimate.

Please, see article:

Percy, D. F., Alkali, B., Generalised proportional intensities models for

repairable systems, IMA Journal of Management Mathematics, 17, 171-185, 2006.

Dear Alex, thanks a lot for your answer.

This may not directly answer your question, but at least consider it. I assume you want to know if your raw data fit a particular distribution; in my world it is either normal or lognormal.

The first thing we always do is to calculate the QQ-plots for the original data, and then for the log-transformed data. The straighter of the two regression lines indicates the character of the data. The advantage is that you can see where the data deviate from the model - this is better than just getting a single value like K-S or Shapiro-Wilk test. If data deviate at the low end - you probably have a sensitivity issue and the rest are ok. If they deviate at the high end, you could have saturation (low), or a mixture of two distributions.

We have published this:

Pleil JD, 2016. QQ-plots for assessing distributions of biomarker measurements and generating defensible summary statistics, Journal of Breath Research (Aug.) 10:035001.

Contact me if you'd like: [email protected]

Dear Joachim, thank you very much for your detailed answer. I will carefully study your answer. First comment - I am going to analyse rather Weibull CDF, not normal or log-normal. Second Comment - single points of our statistics are not independent, so we could not use regular (as Chi-Square, Kolmogorov-Smirnov,...) tests. Regards, Sergey.

Dear Protsky,

Please have a look at the attached file.

Dear Mahdi, thanks a lot! I will analyze chapter 2.2 "Fit of distributions by maximum likelihood estimation" with large attention! Regards, Sergey.

Please see below link:

https://cran.r-project.org/web/packages/AdequacyModel/AdequacyModel.pdf

The suggested approach is to compute critical value/p-value by simulation, because when you estimate the parameters from the data rather than using hypothesized values then the distribution of the KS statistic does not follow the null distribution.

You can instead ignore the p-values obtained from the KS test and instead simulate a bunch of datasets from the required distribution (with a meaningful set of parameters) of the same size as your real data. Then, for each set estimate the parameters and do the KS test using the estimated parameters. You p-value will be the proportion of test statistics from the simulated sets that are more extreeme than for your original data.

https://www-cdf.fnal.gov/physics/statistics/notes/cdf1285_KS_test_after_fit.pdf

A correct approach is https://books.google.at/books?hl=en&lr=&id=gHt3j2BxXYkC&oi=fnd&pg=PR5&dq=Distribution+Theory+for+Tests+Based+on+the+Sample+Distribution+Function&ots=KQnvMTyRGF&sig=R3Muhhc3gyy3DFn--tYQcaEj5WA

And

https://www.jstor.org/stable/2334482?seq=1#page_scan_tab_contents

/O-8L

Dear Morad, thanks for your link. I will carefully study it. Regards, Sergey.

Dear Sajid, thanks a lot for your detailed answer. Regards, Sergey.

Dear Fausto, thanks for answer. I fully agree with you, that using MLE we can estimate teta and beta and to calculate CI for them. But my goal is some else - to estimate goodness-of-fit of CDF, i.e. to get numerical estimation, how is proposed CDF is available for input statistics. I don't want to use classical approaches (Chi-square, Kolmogorov-Smirnoff, Mizes, etc.), I want directly use MLE estimations, which I have got during teta and beta calculations. Regards, Sergey

I guess I misunderstood what you wanted. I still maintain that using a single parameter estimate to describe goodness of fit for a range of values is just a first step. I like to see where the deviations from the model actually occur rather than just making a less informed yes/no decision. It doesn't really matter what model you choose as long as you align real-world data with model and observe deviations. There are times in statistics when the subject matter expertise must over-ride blind adherence to a statistics developed by folks who really do not appreciate what you need pragmatically.

Dear Fausto, problem is, that really I don't know, is " OBVIOUSLY if the "real" distribution is the Weibull...". I have some input statistics, it may be Weibull, may be Normak, may be log-Normal. Usually we assume, that it is Weibull. But I should check this supposition by means of goodnes-of-fit test.

Dear Joachim , I fully agree with you, but as first step I really should get some goodness-of-fit estimations. Usually we use classical tests, but for my specific task they are un-available.

Dear Dr Porotsky

You can use some of the standard functions in R such as optimize, nlminb and nlm or use the mle function at stats4 package and maxLik function at maxLik package.

Dear Dr Porotsky

I have a code on R to find the goodness of fit through maximum likelihood estimate.

Dear Salman, unfortunately I didn't use R, I use Matlab. Do you have Matlab code or algorithm description? Regards, Sergey.

Dear Dr Porotsky

I did not use Matlab.

Hi, MLE fits not so well for GoF, but you can simple do MLE for one model, and for another one, like normal and Student-t. If the log-likelihood difference is large, then one model is significantly better than the other. If one model has more parameters, then it fits usually better. However, even if e.g. the true model would be normal, e.g. a Gaussian mix or Student-t would fit BETTER, although the normal model would be preferable in this case. To solve that problem you may use AIC or BIC.

Bye Stephan

Based on Max. Log L values which of the following models is the best fitted model?? And Why only particular...? Plz explain..

Model 2: -131149.63

Model 3: -124778.69

Model 4: -121360.55

Model 5: -119774.63

Model 6: -119052.09

Dear Sergey Porotsky Sir

In Maximum likelihood test, likelihood function can be computed for a number of statistical distributions. The parameters involved are replaced by their respective MLEs based on the given sample. A distribution that has maximum likelihood value or equivalently minimum negative log likelihood value is said to be the most plausible model in that particular situation.

Dear Sergey Porotsky

After choosing a model that can mathematically represent our data we have to estimate parameters of such model. There are several estimate methods in statistical literature, e.g.,

1) Analogic

2) Moments

3) Maximum likelihood

These are two useful articles to fitting distributions dealing shortly with theoretical issues and practical ones using the statistical programming language R, see attached files.

With My Best Wishes.

Dear Ahmed, thanks a lot for your answer and attached articles. Regards, Sergey.