Premium
LIMIT THEOREMS FOR PROPORTIONS OF OBSERVATIONS FALLING INTO RANDOM REGIONS DETERMINED BY ORDER STATISTICS
Author(s) -
Dembińska Anna
Publication year - 2012
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/j.1467-842x.2012.00667.x
Subject(s) - mathematics , order statistic , independent and identically distributed random variables , infinity , statistics , limit (mathematics) , statistic , sample (material) , order (exchange) , population , central limit theorem , random variable , distribution (mathematics) , mathematical analysis , chemistry , demography , finance , chromatography , sociology , economics
Summary In this paper, we study asymptotic behavior of proportions of sample observations that fall into random regions determined by a given Borel set and an order statistic. We show that these proportions converge almost surely to some population quantities as the sample size increases to infinity. We derive our results for independent and identically distributed observations from an arbitrary cumulative distribution function, in particular, we allow samples drawn from discontinuous laws. We also give extensions of these results to the case of randomly indexed samples with some dependence between observations.