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I am seeking a parametric probability model whose pdf has the following characteristics: (1) it is supported on a variate axis that is bounded between 0 and 1; (2) it is continuous; and (3) it is capable of having two (or more) modes that are not necessarily located at 0 or 1 (e.g., in contrast to the beta distribution whose a and b are both <1). I am thinking of some sort expansion on the beta or Kumaraswamy distribution analogous to the bi-Weibull expansion on the Weibull distribution. My intention is to use such a model as a sampling distribution for sample proportions that vary continuously. My simulated data tend toward the low side of the variate range, typically ranging between 0 and 0.01, with a center of mass that in the neighborhood of 0.001. See the illustration below for examples. Has anyone run across such an obscure distribution that meets this challenging combination of criteria?

Results of a Monte Carlo simulation exploring the temporal distribution of 30 random samples.  Four timeslices are shown.  The x axis describes the sample proportion for each single simulation run, the histogram, rug plot, and kernel density estimate describe the distribution of sample proportions for 2000 runs

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  • $\begingroup$ Heya! Welcome and nice first question. Is there any way you can edit the title to give a teeny bit more idea about the content of your question? You can edit the question by clicking the "edit" link in the lower left. $\endgroup$ – Alexis Jul 22 '14 at 21:00
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    $\begingroup$ Title revised, hopefully for the better. $\endgroup$ – UW Will Jul 22 '14 at 21:17
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    $\begingroup$ Why not a mixture of Betas? $\endgroup$ – user44764 Jul 22 '14 at 23:03
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    $\begingroup$ I'd look at a two-component mixture of some continuous family of distributions on the unit interval. (Edit: Uh, like the one Matthew suggested while I was typing.) $\endgroup$ – Glen_b Jul 22 '14 at 23:04
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Why not consider a finite mixture distribution?

In this case, probably one with two components, though some of those plots might just be better approximated by three-component mixtures.

Each component would be something suitable for a continuous proportion.

A typical example would be to use two different beta distributions.

So if the observed proportion is

$$Y=wX_1+(1-w)X_2\,,$$

where the components are say

$$X_i\sim\text{Beta}(\alpha_i,\beta_i),\quad i=1,2$$

then the density is

$$f_Y(y) = wf_{X_1}(y)+(1-w)f_{X_2}(y),\quad 0\leq y\leq 1\,.$$

Here's an example:

enter image description here

The parameters of this example:

$\quad X_1\sim \text{Beta}(1,2500)$

$\quad X_2\sim \text{Beta}(4,3500)$

$\quad w=0.35$

The distribution of $Y$ in a two component mixture of betas has 5 parameters, and is very flexible.

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  • $\begingroup$ +1, although the answer might benefit with suggestions on computation strategies for modeling data in this fashion. I realize it's not in the OP's question, but it would still be helpful. $\endgroup$ – user44764 Jul 23 '14 at 0:12
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I like the finite mixture approach, a lot. Such an approach didn't occur to me at all, because I am not very familiar with finite mixture models. Instead, I was envisioning a multi-component expansion on a beta or Kumaraswamy distribution, analogous to the multi-component models I am familiar with from survival analysis in demography, like the bi-Weibull or Siler distributions. I decided to see if I had it in me to work with the Kumaraswamy distribution, because it has an analytically tractable CDF (unlike the beta distribution):

enter image description here

enter image description here

enter image description here

The challenge here, though, is to fit each component's a and b so that the pdf looks right. Just tinkering with it, I came up with

enter image description here

Based on the R code

d.multi.kumar=function(y,a,b){
    k=length(a)

    p.per.k=rep(0,k)
    for (i in 1:k){
        p.per.k[i]=(1-y^a[i])^b[i]
    }
    S.multi.kumar=prod(p.per.k)

    h.per.k=rep(0,k)
    for (i in 1:k){
        h.per.k[i]=(a[i]*b[i]*y^(a[i]-1))/(1-y^a[i])
    }
    h.multi.kumar=sum(h.per.k)
    return(S.multi.kumar*h.multi.kumar)
}

a=c(1.01,4.2)
b=c(150,1000000000)
y=seq(0,0.2,0.000001)

pdf.multi.kumar=sapply(X=y,FUN=d.multi.kumar,a,b)
plot(y,pdf.multi.kumar,"h",xlim=c(0,0.1),xlab="y",ylab="f(y)",col="light blue")
    abline(h=0)

I like the visual effect of the finite mixture model more. Wrestling sharp and horizontally separated peaks out of a multi-Kumaraswamy distribution seems quite difficult. In terms of first principles, I don't have any great guidance, so I am not sure that a mixture model has any more theoretical warrant than the other approach, but whatever works ...

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