SOME NEW RESULTS ON THE LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER GAMMA MODELS

2015 ◽  
Vol 29 (4) ◽  
pp. 597-621 ◽  
Author(s):  
Peng Zhao ◽  
Yanni Hu ◽  
Yiying Zhang
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Scale Parameters ◽  
General Sufficient Condition ◽  
Weak Majorization ◽  
Majorization Order ◽  
Star Order ◽  
Theoretical Results

In this paper, we carry out stochastic comparisons of the largest order statistics arising from multiple-outlier gamma models with different both shape and scale parameters in the sense of various stochastic orderings including the likelihood ratio order, star order and dispersive order. It is proved, among others, that the weak majorization order between the scale parameter vectors along with the majorization order between the shape parameter vectors imply the likelihood ratio order between the largest order statistics. A quite general sufficient condition for the star order is presented. The new results established here strengthen and generalize some of the results known in the literature. Numerical examples and applications are also provided to explicate the theoretical results.

STOCHASTIC COMPARISONS OF LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER EXPONENTIAL MODELS

2012 ◽  
Vol 26 (2) ◽  
pp. 159-182 ◽  
Author(s):  
Peng Zhao ◽  
N. Balakrishnan
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Exponential Models ◽  
Usual Stochastic Order

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

Likelihood ratio comparisons and logconvexity properties of p-spacings from generalized order statistics

2021 ◽  
pp. 1-20
Author(s):  
Mahdi Alimohammadi ◽  
Maryam Esna-Ashari ◽  
Jorge Navarro
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Record Values ◽  
Generalized Order Statistics ◽  
Model Parameters ◽  
Stochastic Comparisons ◽  
Sequential Order Statistics ◽  
Progressive Type ◽  
Likelihood Ratio Ordering ◽  
Generalized Order

Due to the importance of generalized order statistics (GOS) in many branches of Statistics, a wide interest has been shown in investigating stochastic comparisons of GOS. In this article, we study the likelihood ratio ordering of $p$ -spacings of GOS, establishing some flexible and applicable results. We also settle certain unresolved related problems by providing some useful lemmas. Since we do not impose restrictions on the model parameters (as previous studies did), our findings yield new results for comparison of various useful models of ordered random variables including order statistics, sequential order statistics, $k$ -record values, Pfeifer's record values, and progressive Type-II censored order statistics with arbitrary censoring plans. Some results on preservation of logconvexity properties among spacings are provided as well.

STOCHASTIC MONOTONICITY OF CONDITIONAL ORDER STATISTICS IN MULTIPLE-OUTLIER SCALE POPULATION

2016 ◽  
Vol 31 (3) ◽  
pp. 366-380
Author(s):  
Ebrahim Amini-Seresht ◽  
Yiying Zhang
Keyword(s):  
Distribution Function ◽  
Order Statistics ◽  
Likelihood Ratio ◽  
Monotonicity Property ◽  
Independent Random Variables ◽  
Likelihood Ratio Order ◽  
Stochastic Monotonicity ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Parametric Families

This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let X1, …, Xn be a set of non-negative independent random variables with Xi, i=1, …, p, having common distribution function F(λ1x), and Xj, j=p+1, …, n, having common distribution function F(λ2x), where F(·) denotes the baseline distribution. Let Xi:n(p, q) be the ith smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$. Under the assumptions that xf′(x)/f(x) is decreasing in x∈ℛ+, λ1≤λ2 and s1≤s2, it is shown that $X_{i:n}^{s_{1}}(p+k,q-k)$ is larger than $X_{i:n}^{s_{2}}(p,q)$ according to the likelihood ratio order for any 2≤i≤n and k=1, 2, …, q. Some parametric families of distributions are also provided to illustrate the theoretical results.

Stochastic comparisons of largest-order statistics for proportional reversed hazard rate model and applications

2020 ◽  
Vol 57 (3) ◽  
pp. 832-852
Author(s):  
Lu Li ◽  
Qinyu Wu ◽  
Tiantian Mao
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Parallel Systems ◽  
Rate Model ◽  
Stochastic Comparisons ◽  
Reversed Hazard Rate ◽  
Generalized Gamma ◽  
Pareto Distributions ◽  
Hazard Rate Model

AbstractWe investigate stochastic comparisons of parallel systems (corresponding to the largest-order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate (PRHR) model. As applications of the main results, we obtain the equivalent characterizations of stochastic comparisons with respect to the reversed hazard rate and likelihood rate orders for the exponentiated generalized gamma and exponentiated Pareto distributions. Our results recover and strengthen some recent results in the literature.

ON PARALLEL SYSTEMS WITH HETEROGENEOUS GAMMA COMPONENTS

2011 ◽  
Vol 25 (3) ◽  
pp. 369-391 ◽  
Author(s):  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Random Variables ◽  
Parallel Systems ◽  
Likelihood Ratio Order ◽  
Maximum Order ◽  
Hazard Rate Order

In this article, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous gamma components in terms of the likelihood ratio order and the hazard rate order. LetX1andX2be two independent gamma random variables withXihaving shape parameterr>0 and scale parameter λi,i=1, 2, and letX*1andX*2be another set of independent gamma random variables withX*ihaving shape parameterrand scale parameter λ*i,i=1, 2. Denote byX2:2andX*2:2the corresponding maximum order statistics, respectively. It is proved that, among others, if (λ1, λ2) weakly majorize (λ*1, λ*2), thenX2:2is stochastically greater thanX*2:2in the sense of likelihood ratio order. We also establish, among others, that if 0<r≤1 and (λ1, λ2) isp-larger than (λ*1, λ*2), thenX2:2is stochastically greater thanX*2:2in the sense of hazard rate order. The results derived here strengthen and generalize some of the results known in the literature.

LIKELIHOOD RATIO AND HAZARD RATE ORDERINGS OF THE MAXIMA IN TWO MULTIPLE-OUTLIER GEOMETRIC SAMPLES

2012 ◽  
Vol 26 (3) ◽  
pp. 375-391 ◽  
Author(s):  
Baojun Du ◽  
Peng Zhao ◽  
N. Balakrishnan
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Usual Stochastic Order ◽  
Likelihood Ratio Ordering ◽  
Special Case ◽  
Two Parameter

In this paper, we study some stochastic comparisons of the maxima in two multiple-outlier geometric samples based on the likelihood ratio order, hazard rate order, and usual stochastic order. We establish a sufficient condition on parameter vectors for the likelihood ratio ordering to hold. For the special case whenn= 2, it is proved that thep-larger order between the two parameter vectors is equivalent to the hazard rate order as well as usual stochastic order between the two maxima. Some numerical examples are presented for illustrating the established results.

COMPARISONS OF SERIES AND PARALLEL SYSTEMS WITH HETEROGENEOUS COMPONENTS

2013 ◽  
Vol 28 (1) ◽  
pp. 39-53 ◽  
Author(s):  
Weiyong Ding ◽  
Gaofeng Da ◽  
Xiaohu Li
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Parallel Systems ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Series Systems

This paper carries out stochastic comparisons of series and parallel systems with independent and heterogeneous components in the sense of the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. The main results extend and strengthen the corresponding ones by Misra and Misra [18] and by Ding, Zhang, and Zhao [8]. Meanwhile, the results on the hazard rate order of parallel systems and the reversed hazard order of series systems serve as nice supplements to Theorem 16.B.1 of Boland and Proschan [4] and Theorem 3.2 of Nanda and Shaked [20], respectively.

scholarly journals Stochastic Comparisons of Order Statistics and Spacings: A Review

2012 ◽  
Vol 2012 ◽  
pp. 1-47 ◽  
Author(s):  
Subhash Kochar
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Proportional Hazard ◽  
Rate Model ◽  
Stochastic Comparisons ◽  
Scale Parameters ◽  
Hazard Rate Model ◽  
Recent Developments ◽  
Parent Observations ◽  
Sample Spacings

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.

scholarly journals Auditory Device Voice Activity Detection Based on Statistical Likelihood-Ratio Order Statistics

2020 ◽  
Vol 10 (15) ◽  
pp. 5026
Author(s):  
Seon Man Kim
Keyword(s):  
Statistical Model ◽  
Order Statistics ◽  
Likelihood Ratio ◽  
Error Rate ◽  
Voice Activity Detection ◽  
Activity Detection ◽  
Noisy Environments ◽  
Likelihood Ratio Order ◽  
Voice Activity ◽  
False Rejection

This paper proposes a technique for improving statistical-model-based voice activity detection (VAD) in noisy environments to be applied in an auditory hearing aid. The proposed method is implemented for a uniform polyphase discrete Fourier transform filter bank satisfying an auditory device time latency of 8 ms. The proposed VAD technique provides an online unified framework to overcome the frequent false rejection of the statistical-model-based likelihood-ratio test (LRT) in noisy environments. The method is based on the observation that the sparseness of speech and background noise cause high false-rejection error rates in statistical LRT-based VAD—the false rejection rate increases as the sparseness increases. We demonstrate that the false-rejection error rate can be reduced by incorporating likelihood-ratio order statistics into a conventional LRT VAD. We confirm experimentally that the proposed method relatively reduces the average detection error rate by 15.8% compared to a conventional VAD with only minimal change in the false acceptance probability for three different noise conditions whose signal-to-noise ratio ranges from 0 to 20 dB.

LIKELIHOOD RATIO ORDERING OF THE INSPECTION PARADOX

2004 ◽  
Vol 18 (4) ◽  
pp. 503-510 ◽  
Author(s):  
Taizhong Hu ◽  
Weiwei Zhuang
Keyword(s):  
Likelihood Ratio ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Likelihood Ratio Ordering

In this article, some results on stochastic comparisons of the inspection paradox introduced by Ross [Probability in the Engineering and Informational Sciences 17: 47–51 (2003)] are established in the sense of the likelihood ratio order.

Sign in / Sign up

Export Citation Format

Share Document