Bivariate and multivariate distributions with bimodal marginals

Abstract

We consider simple $n-$ variate distributions, which are the special cases of general elliptically contoured and Kotz type distributions. In particular, we analyze the distributions having joint probability density functions (pdf) defined as functions of $(x_1^2+x_2^2+\cdots +x_n^2)$ allowing easy calculations of different probabilities when using transformations with spherical coordinates. In the case of $n=2,$ we give the various examples of such pdfs whose graphs resemble a bell sunken from the middle. These distributions can be used for modelling data clustered in some areas between concentric circles or ellipses. The easy analytical form of considered distributions make it possible to use them in many applications which require simplicity of calculations. The example of probability density function allowing high correlation is also considered. We also discuss the multivariate conditional ordering of random vectors and compute the structure functions considered in the paper probability density functions. The distributions with bimodal marginals can be used in many areas, such as hydrology, biology, medicine, economics, ecology, physics, and astronomy.

Keywords:

• bimodal distributions
• conditionally ordered random vectors
• Joint probability density function

Acknowledgment

I thank the editor and an anonymous reviewer for valuable comments which resulted in improvement of the presentation of this article.