## Mixtures with a limited number of modal intervals.(English)Zbl 0756.62008

Let $$f$$ be a pdf with discontinuities, if any, of the first kind, and let $$F$$ be its cdf. If $$f$$ is a constant $$c$$ in $$[a,b]$$, $$f(x)\leq c$$ in an open neighborhood of $$[a,b]$$ and $$f(x)<c$$ for at least one $$x$$ in every left (right) neighborhood of $$a$$ ($$b$$) then $$[a,b]$$ is called a modal interval. Any point in it is a mode and if there is only one modal interval, $$f$$ is unimodal. If the support of $$f$$ is an interval and $$f$$ is log-concave, then $$f$$ is strongly unimodal.
The author first obtains necessary and sufficient conditions (NASC’s) for $$f$$ to have at most $$s$$ modal intervals in terms of the number of sign changes (i) to the slope of $$f$$, and (ii) to the second order difference of $$F$$. Next, he obtains NASC’s for each mixture $$g$$ of pdfs $$f$$ in a class $$F_ 0$$ to have at most $$s$$ modal intervals. He shows that each $$g$$ is (strongly) unimodal if each special mixture $$p_ 1f_ 1+p_ 2f_ 2$$, $$f_ i\in F_ 0$$, $$p_ i>0$$, is (strongly) unimodal. He gives NASC’s for these special mixtures to have the stated properties. Some applications and analogous results for discrete distributions are also discussed in the paper.

### MSC:

 62E10 Characterization and structure theory of statistical distributions 26A48 Monotonic functions, generalizations
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