# Multimodality of mixtures of more than two Normal distributions

Let

$$\phi(x;\mu,\sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp \left(- \frac{(x-\mu)^2}{2\sigma^2}\right)$$

denote the Gaussian density function ($\sigma > 0$). Let

$$f(x) = \sum_{i=1}^N p_i \phi(x;\mu_i,\sigma_i)$$

denote an $N$-component mixture of normal densities, where $p_i\ge0$ and $\sum_i p_i = 1$. Depending on the parameters $p_i,\mu_i,\sigma_i$, the mixture density $f(x)$ can be unimodal, or have more than one peak.

Ref. [1] gives necessary and sufficient conditions for a mixture of two $(N=2)$ normal densities to be unimodal. Have there been any generalizations applicable to mixtures of more than 2 densities $(N > 2)$?

If there are no rigid conditions, at least an approximate condition would satisfy me. I am mainly interested in the parameter regime $0 \le \mu \le 100$, $\sigma \approx \mu / 2$, and $0 \le p_i \le 1$ arbitrary. $N$ can be anything between 1 and 100.

References

[1] Robertson, C. A., and J. G. Fryer. “Some Descriptive Properties of Normal Mixtures.” Scandinavian Actuarial Journal 1969, no. 3–4 (January 1, 1969): 137–46. https://doi.org/10.1080/03461238.1969.10404590.

• You're not going to get any useful closed-form answers that are generally valid. For an approximation, it behooves you to specify the ranges of values of $p_i,$ $\mu_i,$ and $\sigma_i$ for which you need an approximation, because even that is going to get very messy. – whuber Apr 11 '18 at 17:43
• @whuber Say $0 \le \mu \le 100$, $\sigma \approx \mu / 2$, and $0 \le p_i \le 1$ arbitrary. – becko Apr 11 '18 at 18:59