scholarly journals Complete convergence for arrays of ratios of order statistics

2019 ◽  
Vol 17 (1) ◽  
pp. 439-451
Author(s):  
Yu Miao ◽  
Huanhuan Ma ◽  
Shoufang Xu ◽  
Andre Adler
Keyword(s):  
Order Statistics ◽  
Order Statistic ◽  
Complete Convergence ◽  
Pareto Distribution ◽  
Random Variables ◽  
Independent Random Variables

Abstract Let {Xn,k, 1 ≤ k ≤ mn, n ≥ 1} be an array of independent random variables from the Pareto distribution. Let Xn(k) be the kth largest order statistic from the nth row of the array and set Rn,in,jn = Xn(jn)/Xn(in) where jn < in. The aim of this paper is to study the complete convergence of the ratios {Rn,in,jn, n ≥ 1}.

scholarly journals Limit theorems for ratios of order statistics from uniform distributions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shoufang Xu ◽  
Changlin Mei ◽  
Yu Miao
Keyword(s):  
Order Statistics ◽  
Limit Theorems ◽  
Complete Convergence ◽  
Large Deviation ◽  
Random Variables ◽  
Independent Random Variables ◽  
Moment Conditions ◽  
Uniform Distributions ◽  
Fixed Sample ◽  
Large Numbers

AbstractLet $\{X_{ni}, 1 \leq i \leq m_{n}, n\geq 1\}${Xni,1≤i≤mn,n≥1} be an array of independent random variables with uniform distribution on $[0, \theta _{n}]$[0,θn], and $\{X_{n(k)}, k=1, 2, \ldots , m_{n}\}${Xn(k),k=1,2,…,mn} be the kth order statistics of the random variables $\{X_{ni}, 1 \leq i \leq m_{n}\}${Xni,1≤i≤mn}. We study the limit properties of ratios $\{R_{nij}=X_{n(j)}/X_{n(i)}, 1\leq i < j \leq m_{n}\}${Rnij=Xn(j)/Xn(i),1≤i<j≤mn} for fixed sample size $m_{n}=m$mn=m based on their moment conditions. For $1=i < j \leq m$1=i<j≤m, we establish the weighted law of large numbers, the complete convergence, and the large deviation principle, and for $2=i < j \leq m$2=i<j≤m, we obtain some classical limit theorems and self-normalized limit theorems.

scholarly journals Estimation of a System Performance in Pareto Distribution with Two Independent Random Variables

2014 ◽  
Vol 14 (21) ◽  
pp. 2854-2856 ◽  
Author(s):  
Medhat Ahmed El Damsesy ◽  
Mohammed Mohammed E ◽  
Ahmed Mohammed El Gazar
Keyword(s):  
System Performance ◽  
Pareto Distribution ◽  
Random Variables ◽  
Independent Random Variables

scholarly journals Complete convergence for weighted sums of pairwise independent random variables

2017 ◽  
Vol 15 (1) ◽  
pp. 467-476
Author(s):  
Li Ge ◽  
Sanyang Liu ◽  
Yu Miao
Keyword(s):  
Rate Of Convergence ◽  
Complete Convergence ◽  
Moving Average ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Pairwise Independent Random Variables ◽  
Moving Average Processes

Abstract In the present paper, we have established the complete convergence for weighted sums of pairwise independent random variables, from which the rate of convergence of moving average processes is deduced.

scholarly journals A Note on Conditioning and Stochastic Domination for Order Statistics

2008 ◽  
Vol 45 (2) ◽  
pp. 575-579 ◽  
Author(s):  
Devdatt Dubhashi ◽  
Olle Häggström
Keyword(s):  
Order Statistics ◽  
Order Statistic ◽  
Conditional Distribution ◽  
Random Variables ◽  
Stochastic Domination

For an order statistic (X1:n,…,Xn:n) of a collection of independent but not necessarily identically distributed random variables, and any i ∈ {1,…,n}, the conditional distribution of (Xi+1:n,…,Xn:n) given Xi:n > s is shown to be stochastically increasing in s. This answers a question by Hu and Xie (2006).

Stochastic ordering of order statistics

1991 ◽  
Vol 28 (2) ◽  
pp. 278-286 ◽  
Author(s):  
A. D. Barbour ◽  
T. Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Order Relation ◽  
Stochastic Ordering ◽  
Rate Function ◽  
Independent Random Variables ◽  
Failure Rate Function ◽  
Exponential Random Variables ◽  
The Law ◽  
Monotone Coupling

If Xi, i = 1, ···, n are independent exponential random variables with parameters λ1, · ··, λ n, and if Yi, i = 1, ···, n are independent exponential random variables with common parameter equal to (λ1 + · ·· + λ n)/n, then there is a monotone coupling of the order statistics X(1), · ··, X(n) and Y(1), · ··, Y(n); that is, it is possible to construct on a common probability space random variables X′i, Y′i, i = 1, ···, n, such that for each i, Y′(i)≦X′(i) a.s., where the law of the X′i (respectively, the Y′i) is the same as the law of the Xi (respectively, the Yi.) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ···, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.

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