# On the conjecture of Kochar and Korwar

TL;DR: This article partially solves a conjecture by Kochar and Korwar (1996) in relation to the normalized spacings of the order statistics of a sample of independent exponential random variables with different scale parameters and proves this conjecture for n=4 for both spacings and normalized Spacings.

About: This article is published in Journal of Multivariate Analysis.The article was published on 2010-05-01 and is currently open access. It has received 11 citations till now.

##### Citations

More filters

••

TL;DR: In this article, the authors review some of the recent developments in the area of stochastic comparisons of order statistics and sample spacings and consider the cases when the parent observations are identically as well as non-identically distributed.

Abstract: We review some of the recent developments in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as nonidentically distributed. But most of the time we will be assuming that the observations are independent. The case of independent exponentials with unequal scale parameters as well as the proportional hazard rate model is discussed in detail.

38 citations

••

TL;DR: In this article, it was shown that under some conditions, one largest order statistic Xλn: n is smaller than another one Xθn : n according to likelihood ratio ordering.

Abstract: Let be independent non negative random variables with , i = 1, …, n, where λi > 0, i = 1, …, n and F is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic Xλn: n is smaller than another one Xθn: n according to likelihood ratio ordering. Furthermore, we apply these results when F is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases.

27 citations

••

TL;DR: In this article, the authors obtained various ordering results for the comparisons of two extreme order statistics from scale models when one set of scale parameters majorizes the other, and applied these results when the baseline distributions are exponentiated Weibull or generalized gamma distributions.

Abstract: Stochastic ordering relations between extreme order statistics from exponential, Weibull and gamma distributions have been studied extensively by many researchers in recent years. In this work, we obtain various ordering results for the comparisons of two extreme order statistics from scale models when one set of scale parameters majorizes the other. The new results obtained here are applied when the baseline distributions are exponentiated Weibull or generalized gamma distributions. In this way, we generalize and extend some results established recently in the literature.

16 citations

••

TL;DR: New results are obtained in the area of stochastic comparisons of simple and normalized spacings from two samples of heterogeneous exponential random variables under which successive spacings are ordered in the likelihood ratio ordering.

10 citations

••

TL;DR: In this paper, it was shown that under some conditions, one largest order statistic is smaller than another one according to likelihood ratio ordering when the distribution is a generalized gamma distribution, which includes Weibull, gamma and exponential random variables.

Abstract: Let $X_{\lambda _{1}},X_{\lambda _{2}},\ldots ,X_{\lambda _{n}}$ be independent nonnegative random variables with $X_{\lambda _{i}}\sim F(\lambda _{i}t)$, $i=1,\ldots ,n$, where $\lambda _{i}>0$, $i=1,\ldots ,n$ and $F$ is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic $X_{n:n}^{\lambda }$ is smaller than another one $X_{n:n}^{\theta }$ according to likelihood ratio ordering. Furthermore, we apply these results when $F$ is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases.

6 citations

### Cites background from "On the conjecture of Kochar and Kor..."

...…exponential distribution with different scale parameters; see for instance, Kochar and Kirmani (1996), Dykstra et al. (1997), Bon and Păltănea (1999), Khaledi and Kochar (2000), Kochar and Xu (2009), Joo and Mi (2010), Torrado et al. (2010), Torrado and Lillo (2013), and the references therein....

[...]

##### References

More filters

•

07 Feb 1996TL;DR: Genesis Basic Distributional Results and Properties Order Statistics and their properties MLEs under Censoring and Truncation and Inference Linear Estimation under CENSoring and Information Reliability Estimation and Applications Inferences under Two-Sample and Multi-Sample Situations Tolerance Limits and Acceptance Sampling Plans Prediction Problems Bayesian Inference and Applications Conditional Inference, Characterizations Goodness-of-Fit Tests Outliers and Some Related Inferential Issues Extensions to Estimation Under Multiple-Outlier Models Selection and Ranking Procedures Record Values Related Distributions and Some

Abstract: Genesis Basic Distributional Results and Properties Order Statistics and Their Properties MLEs under Censoring and Truncation and Inference Linear Estimation under Censoring and Inference Reliability Estimation and Applications Inferences under Two-Sample and Multi-Sample Situations Tolerance Limits and Acceptance Sampling Plans Prediction Problems Bayesian Inference and Applications Conditional Inference and Applications Characterizations Goodness-of-Fit Tests Outliers and Some Related Inferential Issues Extensions to Estimation under Multiple-Outlier Models Selection and Ranking Procedures Record Values Related Distributions and Some Generalizations Mixtures - Models and Applications Bivariate Exponential Distributions Inference for Multivariate Exponential Distributions Optimal Tests in Multivariate Exponential Distributions Accelerated Life Testing with Applications System Reliability and Associated Inference Exponential Regression with Applications Two-Stage and Multi-Stage Estimation Two-Stage and Multi-Stage Tests of Hypotheses Sequential Inference Competing Risks Theory and Identifiability Problems Applications in Survival Analysis Applications in Queueing Theory Exponential Classification and Applications Computer Simulations

361 citations

••

184 citations

01 Feb 1989

TL;DR: Theory of permanents provides an effective tool in dealing with order statistics corresponding to random variables which are independent but possibly non-identically distributed as mentioned in this paper, which is illustrated by giving a characterization of symmetric random variables in terms of order statistics and by generalizing some known recurrence relations.

Abstract: Theory of permanents provides an effective tool in dealing with order statistics corresponding to random variables which are independent but possibly nonidentically distributed. This is illustrated by giving a characterization of symmetric random variables in terms of order statistics and by generalizing some known recurrence relations. It is shown that the distribution function of one or more order statistics can be represented in terms of permanents and this fact combined with the Alexandroff inequality is used to demonstrate the log-concavity of certain sequences. The case of order statistics corresponding to independent exponential random variables is considered and the m.g.f. and moments of an order statistic and those of the range are derived explicitly.

155 citations

••

TL;DR: In this paper, the authors compared a linear combination of order statistics from a distribution F and a corresponding linear combination from G where G-1F is convex and starshaped.

Abstract: : Comparisons are obtained between a linear combination of order statistics from a distribution F and a corresponding linear combination from a distribution G where G-1F is (a) convex, and (b) starshaped. The results have applications in life testing where the underlying distribution has monotone failure rate or monotone failure rate on the average.

142 citations

••

TL;DR: In this article, it was shown that the tests of interval analysis are perfectly analogous to those of Analysis of Variance tests for the normal law variation, and that the published tables of Χ 2, t and z (Fisher (5)) can be used to obtain the 5 and 1 per cent levels of significance for the corresponding tests of the Exponential Theory with appropriate modifications in respect of degrees of freedom.

Abstract: A statistical technique called the technique of “interval”
analysis developed for samples drawn at random from the exponential population. It is shown that the tests of interval analysis are perfectly analogous to those of Analysis of Variance tests for the normal law variation. It
is emphctsized that the published tables of Χ 2 , t and z (Fisher (5)) can be used to obtain the 5 and 1 per cent levels of significance for the corresponding tests of the Exponential Theory with appropriate modifications in respect
of degrees of freedom.

134 citations