Quick question. Anyone able to attribute the following kernel to a known probability distribution (univariate, continous on the real line)? $$ p(x) \propto |x|^a \exp\left(-\frac{1}{2} (x-b)^2 \right), x \in R, \qquad a>0, b \in R$$ My goal is to efficiently draw random numbers from this distribution, so maybe its also worth looking at the kernel of $y = x^2$ with Jacobian $1/(2\sqrt y)$

$$ p(y) \propto y^{(a+1)/2 - 1} \exp\left(-\frac{1}{2} y + b y^{\frac{1}{2}} \right) , y \in (0,\infty),$$

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    $\begingroup$ Could you clarify what you mean by "attribute"? You have written down a well-defined family of probability distributions, at least assuming $a \gt -1.$ $\endgroup$ – whuber Jun 19 at 15:08
  • $\begingroup$ Sorry fo lack of clarity. By "attribute" I mean if this distribution is known. If it is, there might be a efficient random number generator that I can search for. And yes, $a > 0$ in my case, while $b \in R$ $\endgroup$ – Tony Jun 19 at 15:12
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    $\begingroup$ It looks like a sort of generalized gamma but it is a bit odd that the parameter $b$, which is like a scale parameter, only occurs in one of the terms. $\endgroup$ – Sextus Empiricus Jun 19 at 16:13
  • $\begingroup$ Hi, yes its strange. For a=0, it would be a normal, while for b=0, it looks like a gamma. So I was searching for a distribution which nests both as special case. The closest I got was the Amoroso Distribution, but there I would need $b$ to rescale $x$ in both terms $\endgroup$ – Tony Jun 19 at 16:24
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    $\begingroup$ Instead of looking for some existing distribution (which would probably be not well know and not well described) maybe you could phrase it as some compound distribution. That would solve your issue of sampling. In what sort of problem does this occur? That information might help to think of the compound distribution. $\endgroup$ – Sextus Empiricus Jun 19 at 17:58

TL;DR We can develop a uniformly bounded rejection sampler, which will generate a variate from the desired density requiring an expected (worst case) $\approx 4.75$ independent uniform variates. Although the set-up is fairly straightforward/fast, it is non-trivial and this approach may be slow with varying parameters (e.g., Gibbs Sampling).

This is a tricky distribution. As mentioned in the comments, this is nearly the Generalized Gamma distribution (with $p=2$ and $d =a+1$), except for the fact that $b$ is not a true location parameter because it occurs only in the second term. I have been searching for a while now, and am unable to find a reference to this distribution anywhere.

A Uniformly Bounded Rejection Sampler

In this paper by Luc Devroye, a uniformly bounded rejection sampler is constructed for the Generalized Inverse Gaussian distribution, and we can follow a similar approach.

Let me redefine the density (up to a constant) as

$$f(x) = x^{\alpha -1}\exp{\left(-\gamma(x-\mu)^2\right)}, x > 0$$

First step is to prove that the density is log-concave. This can be done by showing that

  1. $f'(x)/f(x)$ is monotone decreasing for $x > 0$
  2. $(\log f(x))'' < 0$ for all $x > 0$.

These properties hold whenever $\alpha > 1$. Next, we note that the mode occurs at $$m = \frac{\mu}{2} + \frac{1}{2\gamma}\sqrt{\gamma\left(2\alpha + \gamma\mu^2 -2\right)}.$$ Define \begin{align*} \phi(x) &= f(m)^{-1}f(x+m) \\ \psi(x) &= \log \phi(x) = (\alpha-1)\log(x+m) - \gamma(x+m-\mu)^2 - \log f(m) \end{align*} so that $\phi(0) = 1$ and $\psi(0) = 0$. We will also need the derivative of $\psi(x)$ $$\psi'(x) = \frac{\alpha-1}{x+m} - 2\gamma(x+m-\mu).$$ Finally, you will need to find $s, t > 0$ such that $\psi(-s) = \psi(t) = -1.$ Newton-Raphson should converge fairly quickly, by iterating $$t_0 > 0, \ t_{n+1} = t_n - \frac{\psi(t_n) + 1}{\psi'(t_n)} \quad\text{and}\quad s_0 < 0, \ s_{n+1} = s_n + \frac{\psi(-s_n) + 1}{\psi'(-s_n)}.$$

The Algorithm

INPUTS: s, t, psi, psi'

Compute p  = 1/psi'(-s)
Compute r  = -1/psi'(t)
Compute t' = t + r*psi(t)
Compute s' = s + p*psi(-s)
Compute q  = t' + s'

   Generate U, V, W ~ U(0, 1)
   if U < q/(q + r + p)           then X = -s' + qV
   elseif U < (q + r)/(q + r + p) then X = t' - r*log(V)
   else                                X = -s' + p*log(V)

   if X > t'      then chi = exp(psi(t) + psi'(t)*(x - t))
   elseif X > -s' then chi = 1
   else                chi = exp(psi(-s) + psi'(-s)*(x + s))

UNTIL log(W) <= psi(X) - log(chi)



This approach has some advantages as well as some important disadvantages. The main advantage is that the algorithm is uniformly bounded.

Theorem. Using the algorithm above, the expected number of iterations required to generate a sample is at most $1.581977\ldots$.

Since three independent uniform variates are required at each iteration, we expect that a draw from $f$ can be generated at the (worst case) cost of generating $\approx 4.75$ uniform variates.

Unfortunately, the set-up is non-trivial. In particular, Newton-Raphson is required to find $s$ and $t$. This approach can be approved by explicitly finding $s, t > 0$ such that $\psi(-s) = \psi(t) = -\rho$ for any $\rho > 0$. I am working on this right now, but have yet to find anything. It is also worth noting that this approach may fail when $\alpha < 1$, which may or may not be a problem depending on the application.

In summary, if you are looking to draw a large number of samples from $f$ for fixed parameters, then this method is robust and efficient. If you are looking for a single draw with varying parameters (e.g., Gibbs Sampling) then the required set-up of this algorithm is a substantial disadvantage.

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