On univariate slash distributions, continuous and discrete

Abstract

In this article, I explore in a unified manner the structure of uniform slash and $\alpha$-slash distributions which, in the continuous case, are defined to be the distributions of Y / U and $Y_\alpha /U^{1/\alpha }$ where Y and $Y_\alpha$ follow any distribution on $\mathbb {R}^+$ and, independently, U is uniform on (0, 1). The parallels with the monotone and $\alpha$-monotone distributions of $Y \times U$ and $Y_\alpha \times U^{1/\alpha }$, respectively, are striking. I also introduce discrete uniform slash and $\alpha$-slash distributions which arise from a notion of negative binomial thinning/fattening. Their specification, although apparently rather different from the continuous case, seems to be a good one because of the close way in which their properties mimic those of the continuous case.

Appendices

Appendix A The continuous case: monotone densities

Consider the density, f, of the distribution of $Z = Y\times U$ where Y follows any distribution with density $h>0$ on $\mathbb {R}^+$ and, independently, $U\sim \mathrm{{U}}(0,1)$. Then, f satisfies

$$\begin{aligned} f(z) = \int _z^\infty \frac{1}{y}h(y)\text {d}y \end{aligned}$$

so that

$$\begin{aligned} h(y) = -\,yf'(y). \end{aligned}$$

Validity of h as a density therefore requires that $f'(z) <0$ for all $z>0$ and hence that f is a monotone decreasing density (on $\mathbb {R}^+$). This is a version of Khintchine’s theorem (Khintchine 1938; Feller 1971). The corresponding c.d.f.’s are related by $H(x) = F(x)-xf(x).$ Relaxing the traditional positivity constraint on h to nonnegativity, gaps in the support of h correspond to constant patches in f.

The distribution of $Z_\alpha = Y_\alpha \times U^{1/\alpha }$ where $Y_\alpha$ follows a distribution with density $h_\alpha \ge 0$ on $\mathbb {R}^+$ and, independently, $U\sim \mathrm{{U}}(0,1)$ is that of an $\alpha$-monotone distribution, in which case $Z_\alpha ^\alpha$ has a monotone density (Olshen and Savage 1970; Dharmadhikari and Joag-Dev 1988; Bertin et al. 1997). The $\alpha$-monotone density $f_\alpha$ satisfies

$$\begin{aligned} f_\alpha (z) = \alpha z^{\alpha -1} \int _z^\infty \frac{1}{y^{\alpha }}\,h_\alpha (y)\text {d}y \end{aligned}$$

so that

$$\begin{aligned} f'_\alpha (z) = \frac{(\alpha -1)}{z}f_\alpha (z)-\frac{\alpha }{z} h_\alpha (z) \end{aligned}$$

and

$$\begin{aligned} \alpha h_\alpha (y) = (\alpha -1 ) f_\alpha (y)-yf'_\alpha (y). \end{aligned}$$

A density $f_\alpha$ is, therefore, $\alpha$-monotone iff

$$\begin{aligned} {(\log f_\alpha )'(z) \le \frac{(\alpha -1)}{z},}\qquad \mathrm{for~all~}z>0. \end{aligned}$$

Here, the inequality is strict except when $h_\alpha (z) =0$. Also, $\alpha H_\alpha (x) = \alpha F_\alpha (x)-xf_\alpha (x).$

Appendix B The discrete case: monotone probability mass functions

On $\mathbb {N}_0$, consider the p.m.f. p, of the distribution of $N=U \circ M$ where M follows any distribution with p.m.f. q and, independently, $U \sim \mathrm{U}(0,1).$ Recall that, for fixed $0<u<1$ and $m=0,1,\ldots$, $u\circ m \sim \mathrm{Bi}(m,u)$. It then follows that the distribution of $U \circ m$ is discrete uniform, $\mathrm{{U}}(0,1,\ldots ,m)$, and, finally, a discrete Khintchine’s theorem states that p is monotone nonincreasing iff $N \sim p$ can be written as

$$\begin{aligned} N|M=m \sim \mathrm{{U}}(0,1,\ldots ,m), \qquad M \sim q. \end{aligned}$$

In fact,

$$\begin{aligned} p(n) =\sum _{m=n}^\infty \frac{q(m)}{m+1}, \qquad q(m) = (m+1)\,\left\{ p(m) -p(m+1)\right\} \end{aligned}$$

(Steutel 1988); see also Jones (2018). In terms of c.d.f.’s, ${Q}(n) =$${P}(n)-(n+1)p(n+1).$

Steutel (1988) went on to discuss discrete $\alpha$-monotonicity which corresponds to replacing U by $U^{1/\alpha }$ above. The distribution of $U^{1/\alpha } \circ m_\alpha$ turns out to be the beta-binomial distribution with parameters $m_\alpha ,$$\alpha$ and 1 on $n=0,1,\ldots ,m_\alpha$. This gives rise to Steutel’s (1988) formulae

$$\begin{aligned} p_\alpha (n) = \alpha \frac{\varGamma (n+\alpha )}{n!} \sum _{m=n}^\infty \frac{m!\,q_\alpha (m)}{\varGamma (m+\alpha +1)} \end{aligned}$$

and

$$\begin{aligned} \alpha q_\alpha (m)= (m+\alpha ) p_\alpha (m) -(m+1)p_\alpha (m+1). \end{aligned}$$

From the latter, it can be concluded that discrete $\alpha$-monotonicity corresponds to p having the property that

$$\begin{aligned} {\frac{p_\alpha (n+1)}{p_\alpha (n)} \le \frac{n+\alpha }{n+1}}, \qquad n=0,1,\ldots \,, \end{aligned}$$

the inequality being strict whenever $q_\alpha (m) >0$. Also, $\alpha {Q}_\alpha (n)= \alpha {P}_\alpha (n)-(n+1)p_\alpha (n+1).$ See Jones (2018) for further discussion.

Appendix C On the shape of the uniform slash Poisson probability mass function

Consider the p.m.f. of the uniform slash Poisson distribution on $n=0,1,\ldots ,$ given by (7). First, $p(0)-p(1) = e^{-\lambda } (2-\lambda )/2 >(=)<0$ as $\lambda <(=)>2.$ Second, $p(1)-p(2) = \lambda e^{-\lambda } \left( 2-\lambda \right) /6 >(=)<0$ as $\lambda <(=)>2$ also. Third, $p(2)-p(3) = \lambda e^{-\lambda } \left( 2+2\lambda -\lambda ^2\right) /24 >(=)<0$ as $\lambda <(=)>1+\sqrt{3}\simeq 2.732.$ And for $n=3,4,\ldots ,$

$$\begin{aligned} p(n)-p(n+1)\propto &\, (n+2) \sum _{j=0}^{n-1} \frac{\lambda ^j}{j!} -n \sum _{j=0}^{n} \frac{\lambda ^j}{j!} = 2 \sum _{j=0}^{n-1} \frac{\lambda ^j}{j!} - \frac{\lambda ^n}{(n-1)!}\\> & {} \left\{ 2(n-1)+2\lambda -\lambda ^2\right\} \frac{\lambda ^{n-2}}{(n-1)!}. \end{aligned}$$

This is positive whenever $\lambda < 1+\sqrt{2n{-}1}$, the upper bound being greater than or equal to $1+\sqrt{5} \simeq 3.236.$ The p.m.f. of the uniform slash Poisson distribution is therefore proven to be decreasing for $0<\lambda <2$, to have $p(0)=p(1)=p(2)$ and then to decrease for $\lambda =2$, and to be unimodal with mode at 2 for $2<\lambda <1+\sqrt{3}$, and with equal modes at 2 and 3 when $\lambda = 1+\sqrt{3}$.

From numerical evidence, I conjecture but cannot prove that p remains unimodal for all larger values of $\lambda$ with its mode, occasionally shared over two consecutive values of n, at or a little greater than $\lambda$.