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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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.
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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