A point-process approach to almost-sure behaviour of record values and order statistics

1996 ◽  
Vol 28 (02) ◽  
pp. 426-462 ◽  
Author(s):  
Charles M. Goldie ◽  
Ross A. Maller
Keyword(s):  
Order Statistics ◽  
Point Process ◽  
Relative Stability ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Record Values ◽  
Process Approach ◽  
Integral Conditions ◽  
Comprehensive Investigation ◽  
Trimmed Sums

Point-process and other techniques are used to make a comprehensive investigation of the almost-sure behaviour of partial maxima(the rth largest among a sample ofni.i.d. random variables), partial record valuesand differences and quotients involving them. In particular, we obtain characterizations of such asymptotic properties asa.s. for some finite constantc, ora.s. for some constantcin [0,∞], which tell us, in various ways, how quickly the sequences increase. These characterizations take the form of integral conditions on the tail ofF,which furthermore characterize such properties as stability and relative stability of the sequence of maxima. We also develop their relation to the large-sample behaviour of trimmed sums, and discuss some statistical applications.

A point-process approach to almost-sure behaviour of record values and order statistics

1996 ◽  
Vol 28 (2) ◽  
pp. 426-462 ◽  
Author(s):  
Charles M. Goldie ◽  
Ross A. Maller
Keyword(s):  
Order Statistics ◽  
Point Process ◽  
Relative Stability ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Record Values ◽  
Process Approach ◽  
Integral Conditions ◽  
Comprehensive Investigation ◽  
Trimmed Sums

Point-process and other techniques are used to make a comprehensive investigation of the almost-sure behaviour of partial maxima (the rth largest among a sample of n i.i.d. random variables), partial record values and differences and quotients involving them. In particular, we obtain characterizations of such asymptotic properties as a.s. for some finite constant c, or a.s. for some constant c in [0,∞], which tell us, in various ways, how quickly the sequences increase. These characterizations take the form of integral conditions on the tail of F, which furthermore characterize such properties as stability and relative stability of the sequence of maxima. We also develop their relation to the large-sample behaviour of trimmed sums, and discuss some statistical applications.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (1) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

Asymptotic Properties of Linear Functions of Order Statistics for Non­Stationary Mixing Processes

1979 ◽  
Vol 28 (1-4) ◽  
pp. 19-36
Author(s):  
David A. Sotres ◽  
Malay Ghosh
Keyword(s):  
Order Statistics ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Linear Functions ◽  
Mixing Processes

For ø­mixing sequences of random variables, under certain conditions, we provide a representation of linear functions of order statistics as sample means pius remainder terms. The remainder terms are shown to converge to zero almost surely at certain rates.

NONNEGATIVITY OF COVARIANCES BETWEEN FUNCTIONS OF ORDERED RANDOM VARIABLES

2007 ◽  
Vol 21 (4) ◽  
pp. 557-577 ◽  
Author(s):  
Taizhong Hu ◽  
Junchao Yao ◽  
Qingshu Lu
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Random Variable ◽  
Record Values ◽  
Poisson Processes ◽  
Generalized Order Statistics ◽  
Geometric Processes ◽  
Unified Method ◽  
Generalized Order

In this article we investigate conditions by a unified method under which the covariances of functions of two adjacent ordered random variables are nonnegative. The main structural results are applied to several kinds of ordered random variable, such as delayed record values, continuous and discrete ℓ∞⩽-spherical order statistics, epoch times of mixed Poisson processes, generalized order statistics, discrete weak record values, and epoch times of modified geometric processes. These applications extend the main results for ordinary order statistics in Qi [28] and for usual record values in Nagaraja and Nevzorov [25].

scholarly journals Smoothing the Hill Estimator

1997 ◽  
Vol 29 (1) ◽  
pp. 271-293 ◽  
Author(s):  
Sidney Resnick ◽  
Cătălin Stărică
Keyword(s):  
Weak Convergence ◽  
Order Statistics ◽  
Empirical Process ◽  
Asymptotic Variance ◽  
Random Variables ◽  
Process Approach ◽  
Hill Estimator ◽  
Direct Result ◽  
The Hill ◽  
Tail Empirical Process

For sequences of i.i.d. random variables whose common tail 1 – F is regularly varying at infinity wtih an unknown index –α < 0, it is well known that the Hill estimator is consistent for α–1 and usually asymptotically normally distributed. However, because the Hill estimator is a function of k = k(n), the number of upper order statistics used and which is only subject to the conditions k →∞, k/n → 0, its use in practice is problematic since there are few reliable guidelines about how to choose k. The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of k decreases and the successful use of the esimator is made less dependent on the choice of k. A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.

SOME NEW RESULTS ON RÉNYI ENTROPY OF RESIDUAL LIFE AND INACTIVITY TIME

2011 ◽  
Vol 25 (2) ◽  
pp. 237-250 ◽  
Author(s):  
Xiaohu Li ◽  
Shuhong Zhang
Keyword(s):  
Order Statistics ◽  
Residual Life ◽  
Random Variables ◽  
Record Values ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Weighted Distributions ◽  
Inactivity Time ◽  
Weighted Random Variables ◽  
Monotonic Properties

This article deals with Rényi entropies for the residual life and the inactivity time. Monotonic properties of the entropy in order statistics, record values, and weighted distributions are investigated, and the comparison on weighted random variables is studied in terms of residual Rényi entropy as well.

scholarly journals Smoothing the Hill Estimator

1997 ◽  
Vol 29 (01) ◽  
pp. 271-293 ◽  
Author(s):  
Sidney Resnick ◽  
Cătălin Stărică
Keyword(s):  
Weak Convergence ◽  
Order Statistics ◽  
Empirical Process ◽  
Asymptotic Variance ◽  
Random Variables ◽  
Process Approach ◽  
Hill Estimator ◽  
Direct Result ◽  
The Hill ◽  
Tail Empirical Process

For sequences of i.i.d. random variables whose common tail 1 – F is regularly varying at infinity wtih an unknown index –α &lt; 0, it is well known that the Hill estimator is consistent for α–1 and usually asymptotically normally distributed. However, because the Hill estimator is a function of k = k(n), the number of upper order statistics used and which is only subject to the conditions k →∞, k/n → 0, its use in practice is problematic since there are few reliable guidelines about how to choose k. The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of k decreases and the successful use of the esimator is made less dependent on the choice of k. A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.

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