# Seeking a continuous, parametric, bimodal sampling distribution for proportions

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?

• 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. – Alexis Jul 22 '14 at 21:00
• Title revised, hopefully for the better. – UW Will Jul 22 '14 at 21:17
• Why not a mixture of Betas? – user44764 Jul 22 '14 at 23:03
• 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.) – Glen_b Jul 22 '14 at 23:04

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:

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.

• +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. – user44764 Jul 23 '14 at 0:12

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):

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

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 ...