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Given a sample of $n$ observations, which are assumed to be $i.i.d.$ and generated from a continuous probability law. Consider the question of estimating the density function $f(x)$. There are two common constraints to satisfy:

  1. $f(x) \geq 0, \forall x$.
  2. $\int f = 1.$

In addition, I want to enforce that the tail of $f(x)$ decay exponentially. That is, now I have a third constraint for $f(x)$. I am not sure whether this question has been investigated in any detail before. If so, could anyone provide some useful references, please? In addition, how to access the goodness of fit for this type of density function estimation, please? Thank you.

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There is a huge literature on density estimation, just search this site or amazon.com or google books. Your additional requirement of an exponential decay of the estimated density might be solved by logspline density estimation. There is an R (CRAN) package logspline, and the method is covered (very briefly) in Venables & Ripley: "Modern Applied Statistics with S" (Springer, 4th edition).

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