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This might sound dumb but if I have $d_i$, where $i=1 \dots n$ observations and I assume they are exponentially distributed, before I use the MLE, should I transform my data to follow an exponential distribution? The reason I ask is because I often see others who normalize their dataset prior to the MLE if they assume their data is normally distributed.

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migrated from math.stackexchange.com Sep 4 '12 at 21:17

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Of course if the data does not follow an exponential you should not fit the mle for the event rate based on an exponential likelihood. But transforming to exponential is not your only option. Perhaps a more general Gamma distribution is appropriate. Use the best estimator for the appropriate family of distributions if you are going to use a parametric approach. Also how easy is it to figure out an appropriate transformation? That is another issue.

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ok thanks! The best then would be to get the mean of the data that will make the data fit under an exponential distribution. So for instance I use in Matlab expfit, get the $\mu$ and apply each observation a $\frac{1}{\mu}\exp{\frac{-d_i}{\mu}}$ transformation – CharlesM Jul 15 '12 at 17:21
No I am saying to try to fit the data to some other parametric family first. Failing that if you have a way to transform to an exponential then you can try that. – Michael Chernick Jul 15 '12 at 17:29
"Transforming to exponential is not your only option": how is it an option at all for MLE?? – whuber♦ Sep 9 '12 at 18:37
@whuber I didn't not say that. I said that if the data does not fit the exponential you should not use the likelihood function based on the exponential to compute the mle. The statement you quoted simply suggested that there sre other approaches that could be better than attempting to transform the distribution to an exponetial. – Michael Chernick Sep 10 '12 at 2:08
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Yes. Your data $\bf must$ follow the distribution under which you will make your estimations or under which you will run you ML estimator. If this is not the case your ML will not be ML for that data. You should transform your data or create a data which follows exponential distribution.

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superb thanks for the clarification! – CharlesM Jul 15 '12 at 17:17
It is my pleasure. – Seyhmus Güngören Jul 15 '12 at 17:34
-1 This is misleading for many reasons, including (1) data almost never exactly follow the posited distribution--and how well they do follow it is routinely checked with goodness of fit tests afterwards--and (2) transforming the data (based on examination of the data) renders most ML output incorrect, especially confidence intervals. – whuber♦ Sep 9 '12 at 18:40
@whuber you are completely right. I had no objections before I posted my answer. The question is not asking a real time application. According to my understand it is asking, to estimate the parameters of an exponential distribution, via simulations. For example to see how the estimator behaves, if MVUE etc. So the question is if I use another data such as the data that I create from a Gaussian distribution do the job as well?I answered and said that it wont do the job and one needs to have data created from exponential distribution.If there is no direct creation then one can transform the data – Seyhmus Güngören Sep 9 '12 at 18:49
@whuber I never talked about what happens in reality. Please have a look at the question once again. I know whatever you are talking about. Do you think that I should have given such a great answer for the needs of this question? – Seyhmus Güngören Sep 9 '12 at 18:51
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