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?

  • $\begingroup$ The total mass usually isn't $1$. $\endgroup$ Jul 31 '12 at 13:22
  • $\begingroup$ 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. $\endgroup$ Jul 31 '12 at 16:06
  • $\begingroup$ 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. $\endgroup$
    – Suvrit
    Jul 31 '12 at 20:48
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    $\begingroup$ 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. $\endgroup$ Aug 1 '12 at 6:15
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    $\begingroup$ 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. $\endgroup$ Aug 1 '12 at 16:43

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...)


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