CDF that combines properties of Pareto and Exponential

Question

Let $Y$ be a random variable defined on the domain $[1;\infty)$ that is distributed according to the cdf $G_Y(y)$.

A Pareto distribution,
$$ G_Y(y) = 1 - y^{-\theta}$$ has the property that $$ P(Y>ay) = 1 - G_Y(ay) = a^{-\theta} (1-G_Y(y)).$$ Similarly, the exponential distribution, $$ G_Y(y) = 1 - \exp(-\theta (y-1)).$$ has the property that $$ P(Y>y+b) = 1- G_Y(y+b) = \exp(b)^{-\theta} (1-G_Y(y)).$$

Is there any distribution that combines these two properties, i.e. where $$ P(Y>ay+b) = z(a,b) (1-G_Y(y)),$$ where $z(a,b)$ is some function of $a$ and $b$?

I expect this distribution to work for any real parameters $a$ or $b$ where $a$ is strictly positive and $b$ can be zero or negative. I expect $z(a,b)$ to be a function of both $a$ and $b$.

The solution should hold for all possible values of Y that are in the domain $[1;\infty)$, and any combination of $a$ and $b$ where $a$ is strictly positive. Of course, if a solution can only found for a subset of these, then that's better than nothing. One might has to restrict $a$ and $b$ such that $aY+b$ has the same domain as $Y$.

Answer 1

The only possible distribution on $[1,\infty)$ satisfying the key equation above of \begin{align} P[&Y>ay+b]=z(a,b)\,P[Y>y]\\ &\text{ whenever } a>0, b\le 0, y\ge 1 \end{align} is the distribution with $P[Y=1]=1$, concentrated entirely at $y=1$.

If the distribution is not concentrated entirely at $Y=1$, then:

• Let $s$ be such that $s>1$ and $P[Y>s]$ is positive.

• Let $t$ be such that $t>s$ and $P[Y>t]<P[Y>s]$.

• Let $r = \sqrt{s}$, so $1<r<s<t$, and $P[Y>r]$ is also positive.

• Solve for $a$ and $b$ such that

\begin{align} ar+b&=r\\ as+b&=t\\ \end{align} Subtracting these two equations gives $a(s-r)=(t-r)$, so $a>1$, and that combined with the first of them gives $b<0$. Now:

The key equation for $y=r$ gives $P[Y>r]=z(a,b)P[Y>r]$, so $z(a,b)=1$.

The key equation for $y=s$ gives $P[Y>t]=z(a,b)P[Y>s]$, so $z(a,b)<1$.

This is a contradiction, so the distribution must have been entirely concentrated at $Y=1$.