Stability theorems for large order statistics with varying ranks

R. J. Tomkins and Hong Wang

Statistics & Probability Letters, 1992, vol. 14, issue 2, 91-95

Abstract: Let {Xn}; be a sequence of i.i.d. random variables with distribution function F. For r [greater-or-equal, slanted] 1 and n [greater-or-equal, slanted] r, let Zrn be the rth largest of };X1,...,Xn};. Let [mu]n = F-1(1 - n-1), n [greater-or-equal, slanted] 1, and suppose that {rn}; is a non-decreasing integer sequence such that 1 [less-than-or-equals, slant] rn} [less-than-or-equals, slant] n. It is shown that Zrnn/[mu]n --> 1 in probability if and only if (*) rn/(n(1 - F([delta][mu]n))) --> 0 for every [delta] 1 a.s. for some r [greater-or-equal, slanted] 1, then Zrnn/[nu]n --> 1 a.s. if and only if (*) holds. These results generalize a theorem of Hall (1979).

Keywords: Relative; stability; almost; sure; stability; large; order; statistics; varying; ranks (search for similar items in EconPapers)
Date: 1992
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