The asymptotic theory of concomitants of order statistics

1974 ◽  
Vol 11 (04) ◽  
pp. 762-770 ◽  
Author(s):  
H. A. David ◽  
J. Galambos
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Random Sample ◽  
Order Statistic ◽  
Asymptotic Theory ◽  
Selection Procedures ◽  
Concomitants Of Order Statistics ◽  
Bivariate Normal ◽  
Independent Identically Distributed

In a random sample of n pairs (X r , Y r ), r = 1, 2, …, n, drawn from a bivariate normal distribution, let Xr :n be the rth order statistic among the Xr and let Y [r:n] be the Y-variate paired with Xr :n . The Y[r:n] , which we call concomitants of the order statistics, arise most naturally in selection procedures based on the Xr :n . It is shown that asymptotically the k quantities k fixed, are independent, identically distributed variates. In addition, putting Rt,n for the number of integers j for which , the asymptotic distribution and all moments of n– 1 Rt, n are determined for t such that t/n → λ with 0 < λ < 1.

The asymptotic theory of concomitants of order statistics

1974 ◽  
Vol 11 (4) ◽  
pp. 762-770 ◽  
Author(s):  
H. A. David ◽  
J. Galambos
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Random Sample ◽  
Order Statistic ◽  
Asymptotic Theory ◽  
Selection Procedures ◽  
Concomitants Of Order Statistics ◽  
Bivariate Normal ◽  
Independent Identically Distributed

In a random sample of n pairs (Xr, Yr), r = 1, 2, …, n, drawn from a bivariate normal distribution, let Xr:n be the rth order statistic among the Xr and let Y[r:n] be the Y-variate paired with Xr:n. The Y[r:n], which we call concomitants of the order statistics, arise most naturally in selection procedures based on the Xr:n. It is shown that asymptotically the k quantities k fixed, are independent, identically distributed variates. In addition, putting Rt,n for the number of integers j for which , the asymptotic distribution and all moments of n–1Rt, n are determined for t such that t/n → λ with 0 < λ < 1.

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (03) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

Let X (1) ≦ X (2) ≦ ·· ·≦ X (N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt /t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (3) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

LetX(1)≦X(2)≦ ·· ·≦X(N(t))be the order statistics of the firstN(t) elements from a sequence of independent identically distributed random variables, where {N(t);t≧ 0} is a renewal counting process independent of the sequence ofX's. We give a complete description of the asymptotic distribution of sums made from the topktextreme values, for any sequencektsuch thatkt→ ∞,kt/t→ 0 ast→ ∞. We discuss applications to reinsurance policies based on large claims.

scholarly journals OPTIMAL MOMENT INEQUALITIES FOR ORDER STATISTICS FROM NONNEGATIVE RANDOM VARIABLES

2019 ◽  
pp. 1-15
Author(s):  
Nickos Papadatos
Keyword(s):  
Order Statistics ◽  
Order Statistic ◽  
Random Variables ◽  
Upper Bounds ◽  
Moment Inequalities ◽  
Population Mean ◽  
Sample Minimum ◽  
Independent Identically Distributed

We obtain the best possible upper bounds for the moments of a single-order statistic from independent, nonnegative random variables, in terms of the population mean. The main result covers the independent identically distributed case. Furthermore, the case of the sample minimum for merely independent (not necessarily identically distributed) random variables is treated in detail.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals On dynamic mutual information for bivariate lifetimes

2015 ◽  
Vol 47 (4) ◽  
pp. 1157-1174 ◽  
Author(s):  
Jafar Ahmadi ◽  
Antonio Di Crescenzo ◽  
Maria Longobardi
Keyword(s):  
Mutual Information ◽  
Order Statistics ◽  
Random Sample ◽  
Exponential Model ◽  
Multicomponent Systems ◽  
Distribution Free ◽  
Lifetime Distributions ◽  
Different Ages ◽  
Special Case ◽  
Residual Lifetimes

We consider dynamic versions of the mutual information of lifetime distributions, with a focus on past lifetimes, residual lifetimes, and mixed lifetimes evaluated at different instants. This allows us to study multicomponent systems, by measuring the dependence in conditional lifetimes of two components having possibly different ages. We provide some bounds, and investigate the mutual information of residual lifetimes within the time-transformed exponential model (under both the assumptions of unbounded and truncated lifetimes). Moreover, with reference to the order statistics of a random sample, we evaluate explicitly the mutual information between the minimum and the maximum, conditional on inspection at different times, and show that it is distribution-free in a special case. Finally, we develop a copula-based approach aiming to express the dynamic mutual information for past and residual bivariate lifetimes in an alternative way.

Choice, Order Statistics and the Distribution of Earnings

1997 ◽  
Author(s):  
Michael Sattinger
Keyword(s):  
Order Statistics ◽  
Order Statistic ◽  
Occupational Choice ◽  
Earnings Inequality ◽  
Limiting Distribution ◽  
Asymptotic Results ◽  
Expected Earnings

This paper analyzes the distribution of earnings as being generated by workers choosing among occupations on the basis of earnings maximization. A worker’s earnings then have characteristics of an order statistic. The extension to multiple occupations leads to the revision results from A.D. Roy’s two-occupation case. An additional occupation raises expected earnings while in general reducing earnings inequality. Asymptotic results from order statistics suggest that the process of occupational choice determines a limiting distribution of earnings independently of underlying distributions of occupational abilities.

scholarly journals Higher Order Moments of the Order Statistics for the Rectangular, Exponential, Gamma and Weibull Distributions

2021 ◽  
Vol 2 (3) ◽  
pp. 61-76
Author(s):  
Sampath Kumar ◽  
V. V. HaraGopal
Keyword(s):  
Image Analysis ◽  
Order Statistics ◽  
Order Statistic ◽  
Higher Order ◽  
Weibull Distributions ◽  
Higher Order Moments

In this paper we discuss the problem of Higher Order Moments for the order Statistics for the Rectangular, Exponential, Gamma and Weibull distributions by finding the order statistic distributions for the base distribution and modified distributions, the base distribution is to deduce the corresponding distribution by the polynomial modifier. These higher order moments are very much useful in most of the Data sciences and Image analysis.  

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