4
$\begingroup$

The following probability density function has a particularly simple form, and it was produced when deriving a confidence interval for $\frac{\mu}{\sigma^2}$ , $$g(y;\delta)=c_\delta(1-y^2)^{(n-3)/2}e^{\delta y}\quad\quad \text{for}\quad-1\leq y\leq 1 $$ where $$c_\delta ={{(|\delta |/2)^{n/2-1}}\over{\sqrt{\pi}\, \Gamma((n-1)/2)I_{n/2-1}(|\delta|)}} $$ and where $I_{n/2-1}(|\delta|)$is the modified Bessel function of the first kind of order $n/2-1$. For a description of the context and derivation of the $g(y;\delta)$ density from the Normal distribution, see Bornholt (2022) (pp. 26, 37).

My question is, does this probability density function have a name already? If not, has anyone seen it used in other settings?

$\endgroup$
4
  • 2
    $\begingroup$ This is not a generalised Beta... $\endgroup$
    – Xi'an
    Commented Apr 13, 2022 at 9:08
  • 1
    $\begingroup$ I've not seen this distribution before, but it is quite an interesting one. Would you mind elaborating on the context in which it was derived? $\endgroup$
    – Ben
    Commented Apr 15, 2022 at 2:18
  • $\begingroup$ This density came up in my research that involves complementing hypothesis test results with confidences in the competing hypotheses (based on the sample evidence alone). For one example, I needed a confidence interval for $\frac{\mu}{\sigma^2}$ for n i.i.d. $X_i\sim N(\mu,\sigma^2)$. From a suggestion in the Cox and Hinkley (1974) text, the method is based on the conditional distribution of $Y=\bar X(n/\sum_iX_i^2)^{1/2}$ given $Y_2=y_2$, where $Y_2=\sum_iX_i^2$. The $\delta$ in the above density is defined by $\delta=\sqrt{ny_2}\mu/{\sigma^2}$. $\endgroup$ Commented Apr 15, 2022 at 13:28
  • $\begingroup$ For a fuller description of the context and derivation of the $g(y;\delta)$ density from the Normal distribution, see pages 26 and 37 of papers.ssrn.com/sol3/papers.cfm?abstract_id=3973254 $\endgroup$ Commented Apr 15, 2022 at 23:53

1 Answer 1

3
$\begingroup$

I don't know a name for this distribution, but here is some information that might (or might not) be useful

I have not seen this density function before; a quick search from me did not yield a name. However, it looks to me like this density might be something that emerges from some kind of monotonic density transform, so I'm going to explore that to see if anything useful comes up. (You haven't specified your parameter ranges, so I'm going to assume that $\delta>0$ .) Taking $a \equiv (n-3)/2$ for purposes of simplification, your density function is proportionate to the kernel:

$$g(y) \propto (1-y^2)^a \exp(\delta y) \cdot \mathbb{I}(-1 \leqslant y \leqslant 1).$$

Suppose we now investigate this density only over its support so that we can omit the indicator term. We will try to find a transformation that is monotonic over this support and gives some simplified form. Below I will show two transformations that are interesting (hat tip to JimB in comments for the second one).


Sinusoidal transformation: Taking $y = \sin(x)$ we have $dy = \cos(x) \ dx$ which then gives:

$$\begin{align} g(y) \ dy &\propto (1-\sin^2(x))^a \exp(\delta \sin(x)) \ dy \\[12pt] &= \cos^2(x)^a \exp(\delta \sin(x)) \ dy \\[12pt] &= \cos^{2a}(x) \exp(\delta \sin(x)) \ dy \\[12pt] &= \cos^{n-3}(x) \exp(\delta \sin(x)) \ dy \\[8pt] &= \cos^{n-2}(x) \exp(\delta \sin(x)) \ \frac{dy}{\cos(x)} \\[6pt] &= \cos^{n-2}(x) \exp(\delta \sin(x)) \ dx. \\[6pt] \end{align}$$

Consequently, the related random variable $X = \arcsin(Y)$ has the density kernel:

$$f(x) \propto \cos^{n-2}(x) \exp(\delta \sin(x)) \cdot \mathbb{I}(-\tfrac{\pi}{2} \leqslant x \leqslant \tfrac{\pi}{2}).$$

This is also not a density I recognise as any named class, but it is quite an interesting density. In the special case where $n=3$ this latter density has a closed form for its CDF, given by:

$$F(x) = \frac{\exp(\delta \sin(x)) - \exp(-\delta)}{\exp(\delta) - \exp(-\delta)} \quad \quad \quad \quad \quad \text{for } -\tfrac{\pi}{2} \leqslant x \leqslant \tfrac{\pi}{2}.$$

For higher values of $n$ the distribution becomes more complicated and it does not have a closed form CDF.


Hyperbolic-tangent transformation: Taking $y = \tanh(x)$ we have $dy = \text{sech}^2(x) \ dx$ which then gives:

$$\begin{align} g(y) \ dy &\propto (1-\tanh^2(x))^a \exp(\delta \tanh(x)) \ dy \\[12pt] &= \text{sech}^2(x)^a \exp(\delta \tanh(x)) \ dy \\[12pt] &= \text{sech}^{2a}(x) \exp(\delta \tanh(x)) \ dy \\[12pt] &= \text{sech}^{n-3}(x) \exp(\delta \tanh(x)) \ dy \\[8pt] &= \text{sech}^{n-1}(x) \exp(\delta \tanh(x)) \ \frac{dy}{\text{sech}^2(x)} \\[6pt] &= \text{sech}^{n-1}(x) \exp(\delta \tanh(x)) \ dx \\[6pt] \end{align}$$

Consequently, the related random variable $X = \text{arctanh}(Y)$ has the density kernel:

$$f(x) \propto \text{sech}^{n-1}(x) \exp(\delta \tanh(x)).$$

This is also not a density I recognise as any named class, but it is quite an interesting density. In the special case where $n=3$ this latter density has a closed form for its CDF, given by:

$$F(x) = \frac{\exp(\delta \tanh(x)) - \exp(-\delta)}{\exp(\delta) - \exp(-\delta)} \quad \quad \quad \quad \quad \text{for } x \in \mathbb{R}.$$

For higher values of $n$ the distribution becomes more complicated and it does not have a closed form CDF.


Hypergeometric transformation: We can abstract from the above transformations to use a transformation $y = H(x)$ satisfying the nonlinear ordinary differential equation:

$$H'(x) = (1-H(x)^2)^{-a}.$$

The solution to this differential equation uses a hypergeometric function:

$$x = \text{const} + y \cdot {_2}F_1(\tfrac{1}{2}, -a; \tfrac{3}{2}; y).$$

It lets us write:

$$\begin{align} g(y) \ dy &\propto (1-H(x)^2)^a \exp(\delta H(x)) \ dy \\[12pt] &= \frac{1}{H'(x)} \exp(\delta H(x)) \ dy \\[12pt] &= \exp(\delta H(x)) \ dx, \\[12pt] \end{align}$$

which gives:

$$f(x) \propto \exp(\delta H(x)) \cdot \mathbb{I}(x \in \mathscr{X}).$$

This is a complicated distribution, since it involves the hypergeometric function. It is also not a named class of distributions that I'm aware of.

$\endgroup$
2
  • 1
    $\begingroup$ I wonder if $\tanh ^{-1}(y)$ might be a useful transformation as you describe because with n=5 and δ=0, the resulting density is that of the sample correlation coefficient from a bivariate normal with ρ=0 and sample size 6. $\endgroup$
    – JimB
    Commented Apr 15, 2022 at 4:30
  • $\begingroup$ Thanks @Ben and@JimB. The only link I have to named distributions so far is that the n=3 case can be related to one case from the Von Mises-Fisher Distribution, and I think that is what you have found Ben. The connection with the sample correlation coefficient is interesting also. $\endgroup$ Commented Apr 15, 2022 at 13:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.