Limiting distribution of the maximal spacing when the density function admits a positive minimum

Philippe Barbe

Statistics & Probability Letters, 1992, vol. 14, issue 1, 53-60

Abstract: Let X1, X2,... be a sequence of random variables with common distribution F, and let f be the density function of F. Let X1,n [less-than-or-equals, slant] ... [less-than-or-equals, slant] Xn,n be the order statistics of X1,..., Xn and let Mn = max 2 [less-than-or-equals, slant] i [less-than-or-equals, slant] nXi,n - Xi-1,n be the maximal spacing. We assume that f has a positive minimum in x0 and that f(x0 + h) = f(x0) + hrd sgn(h)(1 + o(1)) when h --> 0. We prove that limn-->[infinity]P[nMn [less-than-or-equals, slant] x + an] = exp(-e-[phi]x) where [phi] = f(x0 and an = [phi]-1 log n - [phi]-1r-1 log log n + [phi]-1 log(r-1d-1/r[Gamma](1/r)[phi]1/r).

Keywords: Maximal; spacing; distribution; function; of; the; spacings (search for similar items in EconPapers)
Date: 1992
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