Order Statistics of Bivariate Exponential Random Variables

1996 ◽  
pp. 129-141 ◽  
Author(s):  
H. N. Nagaraja ◽  
Geraldine E. Baggs
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Bivariate Exponential ◽  
Exponential Random Variables

COMMENTS ON THE SURVEY BY BALAKRISHNAN AND ZHAO

2013 ◽  
Vol 27 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Negative Binomial ◽  
Random Variables ◽  
Hazard Rates ◽  
Proportional Hazard ◽  
Stochastic Comparisons ◽  
Exponential Random Variables ◽  
Binomial Random Variables

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

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.

scholarly journals DEPENDENCE AMONG SPACINGS

2000 ◽  
Vol 14 (4) ◽  
pp. 461-472 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Joint Density ◽  
Multiple Outliers ◽  
Exchangeable Random Variables ◽  
Exponential Random Variables ◽  
Dependence Properties

In this paper, we study the dependence properties of spacings. It is proved that if X1,..., Xn are exchangeable random variables which are TP2 in pairs and their joint density is log-convex in each argument, then the spacings are MTP2 dependent. Next, we consider the case of independent but nonhomogeneous exponential random variables. It is shown that in this case, in general, the spacings are not MTP2 dependent. However, in the case of a single outlier when all except one parameters are equal, the spacings are shown to be MTP2 dependent and, hence, they are associated. A consequence of this result is that in this case, the variances of the order statistics are increasing. It is also proved that in the case of the multiple-outliers model, all consecutive pairs of spacings are TP2 dependent.

THE TOTAL TIME ON TEST STATISTICS AND l1-ISOTROPY

2000 ◽  
Vol 07 (02) ◽  
pp. 143-151 ◽  
Author(s):  
YU HAYAKAWA
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Test Statistics ◽  
Exponential Random Variables ◽  
Characterization Property ◽  
Mutually Independent ◽  
Special Case

In the literature on the total time on test statistics, it is often assumed that the random variables are mutually independent. It is well known that the scaled total time on test statistics of i.i.d. exponential random variables are the order statistics of independent uniform random variables on (0, 1). We show that this is in fact a characterization property of the l1-isotropic sequence of random variables, which includes the sequence of i.i.d. exponential random variables as a special case.

EQUIVALENT CHARACTERIZATIONS ON ORDERINGS OF ORDER STATISTICS AND SAMPLE RANGES

2010 ◽  
Vol 24 (2) ◽  
pp. 245-262 ◽  
Author(s):  
Tiantian Mao ◽  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
The Other ◽  
Stochastic Orders ◽  
Exponential Random Variables ◽  
Discrete Counterpart ◽  
Similarities And Differences ◽  
Geometric Variables

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

Probability Density Function of the Product and Quotient of Two Correlated Exponential Random Variables

1986 ◽  
Vol 29 (4) ◽  
pp. 413-418 ◽  
Author(s):  
Henrick J. Malik ◽  
Roger Trudel
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Exponential Distribution ◽  
Random Variables ◽  
Type I ◽  
Type Ii ◽  
Exponential Distributions ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential ◽  
Exponential Random Variables

AbstractThis article deals with the distributions of the product and the quotient of two correlated exponential random variables. We consider here three types of bivariate exponential distributions: Marshall-Olkin's bivariate exponential distribution, Gumbel's Type I bivariate exponential distribution, and Gumbel's Type II bivariate exponential distribution.

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