# What is the name of this continuous univariate distribution?

I am looking for the (or a) suitable name of a continuous univariate distribution depending on two parameters $\alpha>0$ (shape) and $\beta >0$ (scale). The support is $(0, \,\beta)$ and the survival and density functions are $$S_X(x) = \left[1- x/\beta\right]^\alpha \qquad f_X(x) = \frac{\alpha}{\beta}\, \left[1- x/\beta\right]^{\alpha-1}$$ This is the special case of the Generalised Pareto Distribution (GPD) with location at zero and finite upper end-point $S_{\mathrm{GPD}}(x) = \left[1 + \xi \,x/\sigma \right]^{-1/\xi}$ with $\xi <0$, but here reparametrised through $\alpha := -1/\xi$ and $\beta:=-\sigma/\xi$. Thus the exponential distribution $\xi = 0$ is the limit $\beta \rightarrow +\infty$.

In Wikipedia's list of probability distributions, this is but a special case of Beta (rescaled) and Kumaraswamy's (rescaled).

A comparable reparametrisation for the GPD with $\xi >0$, namely $\alpha := 1/\xi$ and $\beta:=\sigma/\xi$, leads to the Lomax distribution, so a cheap pun would suggest to use maxlo'' distribution.

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"In Wikipedia's list of probability distributions, this is but a special case of Beta (rescaled) and Kumaraswamy's (rescaled)." - isn't this your answer? –  Macro Jun 27 '12 at 13:01
@Macro: my motivation is to find a nice name for a R implementation of this specific 2-parameters distribution. It is only a very special case within the two families and it would be misleading to call it, say, rescBeta. –  Yves Jun 27 '12 at 13:37
(+1 for @Macro) I do not understand the question. In R, this is simply beta*rbeta(n,1,alpha) if you need to simulate it. If you need to draw inference on the parameters, you can call the function whatever you like... –  Xi'an Jun 27 '12 at 18:22
@Xi'an Of course you are right: what is in a name? Yet although the exponential is only a special case of gamma, it is useful to have it in R under this name. –  Yves Jun 28 '12 at 6:12