On the relationships between copulas of order statistics and marginal distributions

Abstract As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T 1, ..., Tr. In any case, we assume that T 1, ..., Tr are identically distributed, with a common survival function ̄G and their survival copula is denoted by K. The diagonal sections of K, along with ̄G, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T 1, ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results.

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An extremal markovian sequence

Journal of Applied Probability ◽

10.1017/s0021900200027236 ◽

1989 ◽

Vol 26 (02) ◽

pp. 219-232 ◽

Cited By ~ 6

Author(s):

M. Teresa Alpuim

Keyword(s):

Distribution Function ◽

Order Statistics ◽

Stationary Distribution ◽

Random Variable ◽

Stationary Sequence ◽

Left Endpoint ◽

Limit Laws ◽

Marginal Distributions ◽

Common Distribution ◽

Common Distribution Function

In this paper we consider an independent and identically distributed sequence {Yn } with common distribution function F(x) and a random variable X 0, independent of the Yi 's, and define a Markovian sequence {Xn } as Xi = X 0, if i = 0, Xi = k max{Xi − 1, Yi }, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex ) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

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Marginal distributions of sequential and generalized order statistics

Metrika ◽

10.1007/s001840300268 ◽

2003 ◽

Vol 58 (3) ◽

pp. 293-310 ◽

Cited By ~ 131

Author(s):

Erhard Cramer ◽

Udo Kamps

Keyword(s):

Order Statistics ◽

Generalized Order Statistics ◽

Marginal Distributions ◽

Generalized Order

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An extremal markovian sequence

Journal of Applied Probability ◽

10.2307/3214030 ◽

1989 ◽

Vol 26 (2) ◽

pp. 219-232 ◽

Cited By ~ 44

Author(s):

M. Teresa Alpuim

Keyword(s):

Distribution Function ◽

Order Statistics ◽

Stationary Distribution ◽

Random Variable ◽

Stationary Sequence ◽

Left Endpoint ◽

Limit Laws ◽

Marginal Distributions ◽

Common Distribution ◽

Common Distribution Function

In this paper we consider an independent and identically distributed sequence {Yn} with common distribution function F(x) and a random variable X0, independent of the Yi's, and define a Markovian sequence {Xn} as Xi = X0, if i = 0, Xi = k max{Xi− 1, Yi}, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

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