Properties of Modal Intervals

Abstract

Let X be a random variable with distribution function F. Define for $x \in E^1 $ and $\beta \in ( {0,1} )$ the function $S( {x;\beta ,F} )$ as the infimum of the points z satisfying\[F( {x + S( {x;\beta ,F} )} ) - F( {x - S( {x;\beta ,F} )} )\geqq \beta .\] In an earlier paper some of the properties of S were established. This paper examines other aspects of S. A relationship is derived between points of discontinuity of F and points of nondifferentiability of S. Under some weak conditions S is shown to be continuous even if F has discontinuities of the first kind. The function S is also related to quantiles and estimation in a parametric setting.