Limit Theorems for Randomly Selected Ratios of Order Statistics from a Pareto Distribution

Consider independent and identically distributed random variables{Xnk, 1≤k≤m, n≥1}from the Pareto distribution. We randomly select two adjacent order statistics from each row,Xn(i)andXn(i+1), where1≤i≤m−1. Then, we test to see whether or not strong and weak laws of large numbers with nonzero limits for weighted sums of the random variablesXn(i+1)/Xn(i)exist, where we place a prior distribution on the selection of each of these possible pairs of order statistics.

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Stochastic comparison of parallel systems with Pareto components

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964821000176 ◽

2021 ◽

pp. 1-13

Author(s):

Sameen Naqvi ◽

Weiyong Ding ◽

Peng Zhao

Keyword(s):

Order Statistics ◽

Extreme Value Theory ◽

Pareto Distribution ◽

Parallel Systems ◽

Value Theory ◽

Extreme Value ◽

Stochastic Comparison ◽

Shape Parameters ◽

Scale Parameters ◽

Dispersive Ordering

Abstract Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$ , $a > 0$ . As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

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Limit Theorems for Order Statistics with Variable Rank Under Exponential Normalization

Sankhya A ◽

10.1007/s13171-021-00275-y ◽

2022 ◽

Author(s):

H. M. Barakat ◽

A. R. Omar

Keyword(s):

Order Statistics ◽

Limit Theorems

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Limit theorems for sums of order statistics

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete ◽

10.1007/bf00534780 ◽

1976 ◽

Vol 33 (4) ◽

pp. 285-307

Author(s):

B. L. S. Prakasa Rao

Keyword(s):

Order Statistics ◽

Limit Theorems

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Representations and limit theorems for extreme value distributions

Journal of Applied Probability ◽

10.1017/s0021900200046027 ◽

1978 ◽

Vol 15 (03) ◽

pp. 639-644 ◽

Cited By ~ 2

Author(s):

Peter Hall

Keyword(s):

Stochastic Process ◽

Order Statistics ◽

Joint Distribution ◽

Limit Theorems ◽

Random Variables ◽

Canonical Representation ◽

Extreme Value ◽

Limiting Distribution ◽

Extreme Value Distributions ◽

Independent Identically Distributed

LetXn1≦Xn2≦ ··· ≦Xnndenote the order statistics from a sample ofnindependent, identically distributed random variables, and suppose that the variablesXnn, Xn,n–1, ···, when suitably normalized, have a non-trivial limiting joint distributionξ1,ξ2, ···, asn → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn,n≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems forξnasn →∞.

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