Generating Minimum Dispersion Densities from an Interval-Valued ...

[a n ; b n ] subintervals of [R; L], with a 1 = b 1 = R, anda n = b n = L.)3) Find the intersection of all distinct pairs of intervals from[a i ; b i ], i = 2; 3; :::; n 1, such that these intersectionsare proper closed intervals; that is, are of the form [a; b]with a < b. Let I be the set of these intersectionstogether with all the proper intervals [a i ; b i ]: (The intersectionof any nonempty family S of the intervals[a i ; b i ] is either a member of S or can be obtained asthe intersection of two members of S.)4) For each interval I 2 I, form the family S I of allintervals [a i ; b i ] containing I, so for each intersection[a; b], S [a;b] is the family of all the intervals in f[a i ; b i ] :i = 2; 3; :::; n 1g containing [a; b]. Note that S [a;b] hasa multiplicity, namely the number of (not necessarilydistinct) intervals containing [a; b]:5) For those intervals [a i ; b i ] not in S I and such that a i 0:268 082 > 1 n = 0:25:There are no more intervals to check, so the minimumis H (x) = 0:268 082, which occurs for x =(0:33; 0:648 305; 0:69; 0:75).From the two preceding examples, it is easy to see whathappens for any family of mutually disjoint intervals: for each