To what family of densities does $e^{-u}(u^k-k!) \log u$ belong?

Question

Apparently $\int_0^{\infty} e^{-u} (u-1) \log u du = 1$, $\int_0^{\infty} e^{-u} \frac{1}{3}(u^2-2) \log u du = 1$ etc. Does these densities belong to a known family of densities? The closest I found were gamma and chisquared distribution, but they do not have a log inside.

Answer 1

The normalization constant needed to make $$\int_{u=0}^\infty f(u \mid k) \, du = 1$$ is $\frac{1}{k! H_k}$ where $H_k = \sum_{m=1}^k \frac{1}{m}$ is the $k^{\rm th}$ harmonic number. Thus we require $$f(u \mid k) = \frac{e^{-u} (u^k - k!) \log u}{k! H_k},\quad u > 0.$$

However, there is a more fundamental problem: for $k > 1$, $f(u \mid k) < 0$ for $u$ in a subset of the support $(0,\infty)$. So it cannot be a density even if normalized. The only case where $f \ge 0$ is when $k = 1$.

We can see this by noting that if $u > 1$ and $k > 1$, there is an interval $$1 < u < (k!)^{1/k}$$ for which $f < 0$. I suppose you could define a proper density by making the support equal to $(k!^{1/k}, \infty)$, but then $f$ will not resemble anything like a well-known distribution.