Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 1, 74-87

STOCHASTIC PROPERTIES OF ORDER STATISTICS FROM FINITE POPULATIONS

By

SUBHASH C. KOCHAR, Indian Statistical Institute, New Delhi

And

RAMESH KORWAR, University of Massachusetts, Amherst

SUMMARY. We study stochastic orders and dependence relations between order statistics from a linearly ordered finite population when using either simple random sampling without replacement (SRSWOR) or Midzuno sampling schemes. It is shown that when there are no multiplicities in the population, the density functions of order statistics, in the cases of SRSWOR and a special case of Midzuno sampling, are logconcave and hence they have increasing failure rate (IFR) distributions. Also in this case the successive order statistics are likelihood ratio ordered. It is also seen that whereas any pair of order statistics is positively quadrant dependent under Midzuno sampling, it may not satisfy many of the stronger notions of positive dependence like positive regression dependence and TP2 dependence etc. However, we able to prove that X(j) for any j > 1. We further discuss some unresolved problems in this area.

AMS (1991) subject classification. 62G30, 60E15, 62D05

Key words and phrases. Simple random sampling without replacement, Midzuno sampling, logconcave density, likelihood ratio ordering, failure rate ordering, T$P\2$ dependence, RTI property, positive quadrant dependence.

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