Welcome to MathOverflow

Question

1

The density of a wrapped normal distribution is given by $$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$ Considering two density functions $f(x),\ g(x)$ of a wrapped normal distribution with respective parameters $\mu_1,\ \mu_2$ and $\sigma_1,\ \sigma_2$, is the product $h(x)=f(x)g(x)$ a density function of a wrapped normal distribution?

edited Aug 1 at 8:12

asked Jul 31 at 13:10

ostap bender
63●4

The total mass usually isn't $1$. – Douglas Zare Jul 31 at 13:22

Yes, it isn't. For the classical case of a normal distribution, it is possible to derive a formula for the new mass. I wonder whether there is a similar formula for the Wrapped Normal case. – ostap bender Jul 31 at 16:06

one way to check is by seeing if the product of two Jacobi Theta functions is again some kind of Jacobi theta function with "reasonable" parameters---might be good to tag this question with "special-functions" to attract the attention of special function experts. – suVRit Jul 31 at 20:48

3

If $\mu_1 = 0$ and $\mu_2 = \pi$ and $\sigma_1=\sigma_2$ then for generic values of $\sigma_1$ the product is bimodal, which I think can't happen for a wrapped normal density. – Douglas Zare Aug 1 at 6:15

1

Douglas' comment answers my question. Bimodality indeed can not happen in a wrapped normal distribution. Thus, the product of two wrapped normal densities is unfortunately not wrapped normal. – ostap bender Aug 1 at 16:43

Igor Rivin · Answer 1 · 2012-08-01 02:47:11Z UTC

This is really a follow-up on Suvrit's comment. There are plenty of formulas for products of theta functions, many of them found in this Iowa State report. (see particularly page 7). Whether any of them answer the OP's question is for the OP to find out (and tell us...)

Product of densities of a wrapped normal distribution

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.