# MLE estimation of parameters, converting normalized observations to integers and back

I am fitting a model's parameters to grouped data by maximizing the likelihood equation:

$L(\theta)=N!\prod_{i=1}^{G}\frac{p_i(\theta)^{n_i}}{n_i!}$

$\theta$ is the vector of parameters. $n_i$ is the number of observations in group i. That is, $n_i=x_i-x_{i-1}$ where $x_i$ is an individual observation. $N$ is the total number of observations ($N=\sum_{i}n_i$). $G$ is the number of groups.

And $p_i(\theta)=F(x_i;\theta)-F(x_{i-1};\theta)$, where $F(x_i;\theta)$ is the distribution function I am trying to fit to the data.

My problem is that the observations are normalized (they are percentiles). So $0<n_i<1$, when I need integers for the factorials. Multiplying $N$ and $n_i$ by 100 and rounding allows me to run the equation in matlab, but then I am not sure what to do with the output in order to convert it back to 1/100 scale. Appreciate some advice. Thanks.

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