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Motivation

I have a prior on a random variable $X\sim \beta^\prime(\alpha,\beta)$ that I would like to use in JAGS. JAGS does not support the $\beta^\prime$ distribution, but it does support the $F$ distributions, and the F distribution is related to the $\beta^\prime(\alpha,\beta)$ like this:

$$\text{if } X\sim \beta^\prime(\alpha,\beta) \text{ then } X\frac{\alpha}{\beta}\sim F(2\alpha, 2\beta)$$

Question

Is there a transformation $$c,d = f(\alpha, \beta)$$ such that:

$$\text{if } X\sim \beta^\prime(\alpha,\beta)\text{ then } X\sim F(c, d)?$$

What is an appropriate analytical approach or solution to this problem?

Current Approach

My solution is to use simulation. Although sufficient for my application, a formal solution would be more satisfying.

   set.seed(0)
   alpha <- 2
   beta  <- 4
   Y <- rf(100000, 2*alpha, 2*beta) * ( beta / alpha )
   parms <-signif(fitdistr(Y, 'f', start = list(df1=1, df2=2))$estimate,2)

Update whuber's answer states that there is no general transformation.

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1 Answer 1

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I don't quite get it: why can't you just generate an $F_{2\alpha, 2\beta}$ variate and rescale it by $\beta/\alpha$? I also don't see the connection between the code in your current approach and your question, because $Y$ appears to be an $F_{2\alpha, 2\beta}$ variate rescaled by $\alpha/\beta$ rather than $\beta/\alpha$.

Nevertheless, taking the question at face value, if there are parameters $c$ and $d$ corresponding to $\alpha$ and $\beta$, then

  1. The mean of $X$ must be $\beta/\alpha$ times the mean of $F_{2\alpha, 2\beta}$ and it is also equal to the mean of $F_{c,d}$.
  2. The mode of $X$ must be $\beta/\alpha$ times the mode of $F_{2\alpha, 2\beta}$ and it is also the mode of $F_{c,d}$.

    Using standard formulas and solving gives a unique solution

    $$\eqalign{ c = & \frac{2 \left(\alpha^2+\alpha^2 \beta \right)}{\alpha -\alpha \beta + 2 \alpha^2 \beta + 2 \beta ^2-2 \alpha \beta^2} \cr d = & \frac{2 \beta^2}{\alpha -\alpha \beta +\beta^2}. }$$

    We can also equate variances:

  3. The variance of $X$ must equal $(\beta/\alpha)^2$ times the variance of $F_{2\alpha, 2\beta}$ and it must equal the variance of $F_{c,d}$.

Except possibly for special values of $\alpha$ and $\beta$ (most notably, $\alpha = \beta$), this requirement conflicts with the preceding solution. Therefore there does not exist any such transformation in general.

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  • $\begingroup$ thanks for your answer - this is exactly what I was looking for. I have fixed the error in my code (rescaled by $\alpha/\beta$. I could rescale the distribution except that to do so would require adding a special case to code that is generalized to use any distribution that is supported by JAGS. $\endgroup$ Commented Dec 13, 2010 at 21:10
  • $\begingroup$ is it appropriate to assume that X*a/b also has an F distribution? $\endgroup$ Commented Dec 13, 2010 at 23:49
  • $\begingroup$ @David That's what you stipulated! The notation you use (and your sample code) imply alpha and beta are constants. If they are not, then you need to tell us what their distributions are. $\endgroup$
    – whuber
    Commented Dec 14, 2010 at 14:51
  • $\begingroup$ sorry, my previous question was poorly written. Maybe I should have asked: if X~F() is kX~F()? I was wondering if there are general rules about transforming continuous random variables by a constant; it seems that a variable X transformed by k would maintain its shape if its pdf is scaled by k>0, but that this would not hold for k=0 or k<0. (and is this somewhere in my intro probability text by S. Ross? - I cant find it). $\endgroup$ Commented Dec 14, 2010 at 16:11

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