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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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