Bounds for mixtures of order statistics from exponentials and applications

  • Authors:

  • Eugen Pltnea

  • Affiliations:

  • -

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2011

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Abstract

This paper deals with the stochastic comparison of order statistics and their mixtures. For a random sample of size n from an exponential distribution with hazard rate @l, and for 1@?k@?n, let us denote by F"k":"n^(^@l^) the distribution function of the corresponding kth order statistic. Let us consider m random samples of same size n from exponential distributions having respective hazard rates @l"1,...,@l"m. Assume that p"1,...,p"m0, such that @?"i"="1^mp"i=1, and let U and V be two random variables with the distribution functions F"k":"n^(^@l^) and @?"i"="1^mp"iF"k":"n^(^@l^"^i^), respectively. Then, V is greater in the hazard rate order (or the usual stochastic order) than U if and only if @l=@?"i"="1^mp"i@l"i^kk, and V is smaller in the hazard rate order (or the usual stochastic order) than U if and only if @l@?min"1"@?"i"@?"m@l"i, for all k=1,...,n. These properties are used to find the best bounds for the survival functions of order statistics from independent heterogeneous exponential random variables. For the proof, we will use a mixture type representation for the distribution functions of order statistics.