Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 1, pp. 90--95

UNIMODALITY WITH VERTEX AT THE ORIGIN

By

S. S. MITRA, The Pennsylvania State University

SUMMARY. It is well known that a probability distribution is unimodal with vertex at the origin if and only if its characteristic function y can be written as $\psi(t)=t^{-1}\int_{0}^{t} \phi(y)dy$, where $\phi$ is a characteristic function. Assuming that $\phi$ corresponds to a probability density which is continuous everywhere, except possibly at the origin, the author answers a question raised by Pal Medgyessey i.e. if F(x) is (0) unimodal when will [F(x)]*n be also (0) unimodal. As an application of his results, the author considers probability densities of the type

f(x)= xl -1 g(x), 0<x<a , l >0

By imposing some regularity conditions on the function g, it is proved that f(x) possesses properties similar to the gamma distributions in the sense that the n-fold convolution of F is (0) unimodal if and only if $\lambda \leq 1/n$.

AMS (1980) subject classification. 60E07.

Key words and phrases. Unimodality.

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