39 results on '"Maochao Xu"'
Search Results
2. Multivariate dependence among cyber risks based on L-hop propagation
- Author
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Maochao Xu, Gaofeng Da, and Peng Zhao
- Subjects
Statistics and Probability ,Economics and Econometrics ,Multivariate statistics ,business.industry ,Computer science ,Association (object-oriented programming) ,Covariance ,Measure (mathematics) ,Risk analysis (engineering) ,Component (UML) ,Statistics, Probability and Uncertainty ,Hop (telecommunications) ,business ,Risk management - Abstract
Dependence among cyber risks has been an essential and challenging component of risk management. The current study characterizes cyber dependence from both qualitative and quantitative perspectives based on L-hop propagation model. From the qualitative side, it is shown that cyber risks always possess positive association based on the proposed risk propagation model. From the quantitative side, an explicit formula for computing the fundamental dependence measure of covariance is provided for an arbitrary network. In particular, we study the impacts of factors—especially external and internal compromise probabilities, propagation depth, and network topologies—on dependence among cyber risks. We conclude by presenting some examples and applications.
- Published
- 2021
3. Data Breach CAT Bonds: Modeling and Pricing
- Author
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Maochao Xu and Yiying Zhang
- Subjects
Statistics and Probability ,Economics and Econometrics ,Actuarial science ,education ,ComputingMilieux_LEGALASPECTSOFCOMPUTING ,Data breach ,behavioral disciplines and activities ,humanities ,Catastrophe bond ,Work (electrical) ,Business ,Statistics, Probability and Uncertainty ,Extreme risk ,Insurance industry ,health care economics and organizations - Abstract
Data breaches cause millions of dollars in financial losses each year. The insurance industry has been exploring the ways to transfer such extreme risk. In this work, we investigate data breach cat...
- Published
- 2021
4. Statistical modeling of computer malware propagation dynamics in cyberspace
- Author
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Peng Zhao, Taizhong Hu, Xing Fang, Shouhuai Xu, Zijian Fang, and Maochao Xu
- Subjects
Statistics and Probability ,Software_OPERATINGSYSTEMS ,021103 operations research ,Computer science ,0211 other engineering and technologies ,Statistical model ,02 engineering and technology ,Articles ,computer.software_genre ,Computer security ,01 natural sciences ,ComputingMilieux_MANAGEMENTOFCOMPUTINGANDINFORMATIONSYSTEMS ,010104 statistics & probability ,Important research ,Dynamics (music) ,Malware ,0101 mathematics ,Statistics, Probability and Uncertainty ,Cyberspace ,computer ,Cyber threats ,Computer Science::Cryptography and Security - Abstract
Modeling cyber threats, such as the computer malicious software (malware) propagation dynamics in cyberspace, is an important research problem because models can deepen our understanding of dynamical cyber threats. In this paper, we study the statistical modeling of the macro-level evolution of dynamical cyber attacks. Specifically, we propose a Bayesian structural time series approach for modeling the computer malware propagation dynamics in cyberspace. Our model not only possesses the parsimony property (i.e. using few model parameters) but also can provide the predictive distribution of the dynamics by accommodating uncertainty. Our simulation study shows that the proposed model can fit and predict the computer malware propagation dynamics accurately, without requiring to know the information about the underlying attack-defense interaction mechanism and the underlying network topology. We use the model to study the propagation of two particular kinds of computer malware, namely the Conficker and Code Red worms, and show that our model has very satisfactory fitting and prediction accuracies.
- Published
- 2022
5. Modeling Malicious Hacking Data Breach Risks
- Author
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Hong Sun, Maochao Xu, and Peng Zhao
- Subjects
Statistics and Probability ,ComputingMilieux_MANAGEMENTOFCOMPUTINGANDINFORMATIONSYSTEMS ,Economics and Econometrics ,Cyber-Insurance ,ComputingMilieux_LEGALASPECTSOFCOMPUTING ,Business ,Data breach ,Statistics, Probability and Uncertainty ,Computer security ,computer.software_genre ,computer ,Hacker - Abstract
Malicious hacking data breaches cause millions of dollars in financial losses each year, and more companies are seeking cyber insurance coverage. The lack of suitable statistical approaches for sco...
- Published
- 2020
6. Cybersecurity Insurance: Modeling and Pricing
- Author
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Lei Hua and Maochao Xu
- Subjects
Statistics and Probability ,010104 statistics & probability ,Economics and Econometrics ,0103 physical sciences ,Business ,0101 mathematics ,Statistics, Probability and Uncertainty ,010306 general physics ,Computer security ,computer.software_genre ,01 natural sciences ,computer - Abstract
Cybersecurity risk has attracted considerable attention in recent decades. However, the modeling of cybersecurity risk is still in its infancy, mainly because of its unique characteristics. In this...
- Published
- 2019
7. Modeling Multivariate Cyber Risks: Deep Learning Dating Extreme Value Theory
- Author
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Maochao Xu, Xing Fang, Mingyue Zhang Wu, Jinzhu Luo, and Peng Zhao
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Multivariate statistics ,Computer science ,0211 other engineering and technologies ,Machine Learning (stat.ML) ,02 engineering and technology ,Machine learning ,computer.software_genre ,01 natural sciences ,Statistics - Applications ,Domain (software engineering) ,Task (project management) ,010104 statistics & probability ,Statistics - Machine Learning ,Applications (stat.AP) ,0101 mathematics ,Extreme value theory ,ComputingMilieux_MISCELLANEOUS ,021103 operations research ,business.industry ,Deep learning ,Articles ,Heavy-tailed distribution ,Artificial intelligence ,Statistics, Probability and Uncertainty ,High dimensionality ,business ,computer - Abstract
Modeling cyber risks has been an important but challenging task in the domain of cyber security. It is mainly because of the high dimensionality and heavy tails of risk patterns. Those obstacles have hindered the development of statistical modeling of the multivariate cyber risks. In this work, we propose a novel approach for modeling the multivariate cyber risks which relies on the deep learning and extreme value theory. The proposed model not only enjoys the high accurate point predictions via deep learning but also can provide the satisfactory high quantile prediction via extreme value theory. The simulation study shows that the proposed model can model the multivariate cyber risks very well and provide satisfactory prediction performances. The empirical evidence based on real honeypot attack data also shows that the proposed model has very satisfactory prediction performances., 25 pages
- Published
- 2021
8. Modeling multivariate cybersecurity risks
- Author
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Shouhuai Xu, Chen Peng, Maochao Xu, and Taizhong Hu
- Subjects
Statistics and Probability ,021110 strategic, defence & security studies ,Multivariate statistics ,Computer science ,Autoregressive conditional heteroskedasticity ,0211 other engineering and technologies ,02 engineering and technology ,Computer security ,computer.software_genre ,01 natural sciences ,Vine copula ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,computer ,Value at risk - Abstract
Modeling cybersecurity risks is an important, yet challenging, problem. In this paper, we initiate the study of modeling multivariate cybersecurity risks. We develop the first statistical a...
- Published
- 2018
9. A Vine Copula Model for Predicting the Effectiveness of Cyber Defense Early-Warning
- Author
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Lei Hua, Shouhuai Xu, and Maochao Xu
- Subjects
Statistics and Probability ,Value (ethics) ,Structure (mathematical logic) ,021110 strategic, defence & security studies ,Warning system ,Computer science ,Mechanism (biology) ,Applied Mathematics ,0211 other engineering and technologies ,02 engineering and technology ,computer.software_genre ,01 natural sciences ,Measure (mathematics) ,Vine copula ,010104 statistics & probability ,Cyber defense ,Risk analysis (engineering) ,Modeling and Simulation ,Information system ,Data mining ,0101 mathematics ,computer - Abstract
Internet-based computer information systems play critical roles in many aspects of modern society. However, these systems are constantly under cyber attacks that can cause catastrophic consequences. To defend these systems effectively, it is necessary to measure and predict the effectiveness of cyber defense mechanisms. In this article, we investigate how to measure and predict the effectiveness of an important cyber defense mechanism that is known as early-warning. This turns out to be a challenging problem because we must accommodate the dependence among certain four-dimensional time series. In the course of using a dataset to demonstrate the prediction methodology, we discovered a new nonexchangeable and rotationally symmetric dependence structure, which may be of independent value. We propose a new vine copula model to accommodate the newly discovered dependence structure, and show that the new model can predict the effectiveness of early-warning more accurately than the others. We also discus...
- Published
- 2017
10. Modeling and predicting extreme cyber attack rates via marked point processes
- Author
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Shouhuai Xu, Chen Peng, Taizhong Hu, and Maochao Xu
- Subjects
Statistics and Probability ,021110 strategic, defence & security studies ,National security ,Computer science ,business.industry ,Unit of time ,Perspective (graphical) ,0211 other engineering and technologies ,Complex system ,020206 networking & telecommunications ,02 engineering and technology ,Computer security ,computer.software_genre ,Point process ,0202 electrical engineering, electronic engineering, information engineering ,Cyber-attack ,The Internet ,Data mining ,Statistics, Probability and Uncertainty ,Extreme value theory ,business ,computer - Abstract
Cyber attacks have become a problem that is threatening the economy, human privacy, and even national security. Before we can adequately address the problem, we need to have a crystal clear understanding about cyber attacks from various perspectives. This is a challenge because the Internet is a large-scale complex system with humans in the loop. In this paper, we investigate a particular perspective of the problem, namely the extreme value phenomenon that is exhibited by cyber attack rates, which are the numbers of attacks against a system of interest per time unit. It is important to explore this perspective because understanding the statistical properties of extreme cyber attack rates will pave the way for cost-effective, if not optimal, allocation of resources in real-life cyber defense operations. Specifically, we propose modeling and predicting extreme cyber attack rates via marked point processes, while using the Value-at-Risk as a natural measure of intense cyber attacks. The point process...
- Published
- 2016
11. ON THE QUASI-STATIONARY DISTRIBUTION OF SIS MODELS
- Author
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Gaofeng Da, Shouhuai Xu, and Maochao Xu
- Subjects
Statistics and Probability ,010104 statistics & probability ,Stationary distribution ,Computer science ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,02 engineering and technology ,Statistical physics ,0101 mathematics ,Management Science and Operations Research ,Statistics, Probability and Uncertainty ,01 natural sciences ,Industrial and Manufacturing Engineering - Abstract
In this paper, we propose a novel method for constructing upper bounds of the quasi-stationary distribution of SIS processes. Using this method, we obtain an upper bound that is better than the state-of-the-art upper bound. Moreover, we prove that the fixed point map Φ [7] actually preserves the equilibrium reversed hazard rate order under a certain condition. This allows us to further improve the upper bound. Some numerical results are presented to illustrate the results.
- Published
- 2016
12. Discrete Truncated Power‐Law Distributions
- Author
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Maochao Xu, Hong Zhu, and Yingchao Xie
- Subjects
Statistics and Probability ,Mathematical optimization ,Monte Carlo method ,Order statistic ,Sample (statistics) ,010103 numerical & computational mathematics ,01 natural sciences ,Power law ,Upper and lower bounds ,010104 statistics & probability ,symbols.namesake ,Heavy-tailed distribution ,symbols ,Pareto distribution ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Intensity (heat transfer) ,Mathematics - Abstract
Summary Discrete power-law distributions have significant consequences for understanding many phenomena in practice, and have attracted much attention in recent decades. However, in many practical applications, there exists a natural upper bound for the probability tail. In this paper, we develop maximum likelihood estimates for truncated discrete power-law distributions based on the upper order statistics, and large sample properties are mentioned as well. Monte Carlo simulation is carried out to examine the finite sample performance of the estimates. Applications in real cyber attack data and peak gamma-ray intensity of solar flares are highlighted.
- Published
- 2016
13. Analysis of Order Statistics from Distributions with Regularly Varying Tails
- Author
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Maochao Xu, Yu Huang, and Yongling Li
- Subjects
Statistics and Probability ,Order statistic ,Calculus ,Statistical physics ,Mathematics - Abstract
In this article, necessary conditions for comparing order statistics from distributions with regularly varying tails are discussed in terms of various stochastic orders. A necessary and sufficient condition for stochastically comparing tail behaviors of order statistics is derived. The main results generalize and recover some results in Kleiber (2002, 2004). Extensions to coherent systems are mentioned as well.
- Published
- 2014
14. Optimal capital allocation based on the Tail Mean–Variance model
- Author
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Maochao Xu and Tiantian Mao
- Subjects
Statistics and Probability ,Economics and Econometrics ,Multivariate statistics ,Mathematical optimization ,Capital (economics) ,Systematic risk ,Econometrics ,Economics ,Mean variance ,Rule-based system ,Statistics, Probability and Uncertainty ,Elliptical distribution ,Capital allocation line - Abstract
This paper studies capital allocation problems with the aggregate risk exceeding a certain threshold. We propose a novel capital allocation rule based on the Tail Mean–Variance principle. General formulas for the optimal capital allocations are proposed. Explicit formulas for optimal capital allocations are derived for multivariate elliptical distributions. Moreover, we give asymptotic allocation formulas for multivariate regular variation variables. Various numerical examples are given to illustrate the results, and real insurance data is discussed as well.
- Published
- 2013
15. COMMENTS ON 'ORDERING PROPERTIES OF ORDER STATISTICS FROM HETEROGENEOUS POPULATIONS: A REVIEW WITH AN EMPHASIS ON SOME RECENT DEVELOPMENTS'
- Author
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Maochao Xu
- Subjects
Statistics and Probability ,Management science ,Computer science ,Order statistic ,Econometrics ,Management Science and Operations Research ,Statistics, Probability and Uncertainty ,Emphasis (typography) ,Industrial and Manufacturing Engineering - Abstract
Professors Balakrishnan and Zhao have written an excellent survey on the recent developments of stochastic comparisons of order statistics, which cover almost every aspect of ordering properties of order statistics from both continuous and discrete heterogeneous populations. My discussion will be limited to the skewness of order statistics and order statistics from heterogeneous populations with different shape parameters.
- Published
- 2013
16. On the sample ranges from heterogeneous exponential variables
- Author
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Narayanaswamy Balakrishnan and Maochao Xu
- Subjects
Discrete mathematics ,Statistics and Probability ,Stochastic ordering ,Numerical Analysis ,Order statistic ,Excess wealth ordering ,Sample (statistics) ,Star (graph theory) ,Range ,Star ordering ,Spacings ,Exponential function ,Order statistics ,Homogeneous ,Statistics ,Range (statistics) ,Hazard rate ordering ,Statistics, Probability and Uncertainty ,Mathematics ,Dispersive ordering - Abstract
In this paper, the sample range from a heterogeneous exponential sample is shown to be larger than that from a homogeneous exponential sample in the sense of the star ordering. Then, by using this result, some equivalent characterizations of stochastic comparisons of sample ranges with respect to various stochastic orders are established. In this process, two open problems mentioned in Mao and Hu (2010) [16] are solved. The main results established here extend and strengthen several known results in the literature including those of Khaledi and Kochar (2000) [8] , Zhao and Li (2009) [22] and Genest et al. (2009) [7] .
- Published
- 2012
- Full Text
- View/download PDF
17. SOME UNIFIED RESULTS ON COMPARING LINEAR COMBINATIONS OF INDEPENDENT GAMMA RANDOM VARIABLES
- Author
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Subhash C. Kochar and Maochao Xu
- Subjects
Statistics and Probability ,Exchangeable random variables ,Independent and identically distributed random variables ,Stochastic process ,Mathematical statistics ,Management Science and Operations Research ,Star (graph theory) ,Industrial and Manufacturing Engineering ,Combinatorics ,Applied mathematics ,Stochastic optimization ,Statistics, Probability and Uncertainty ,Linear combination ,Random variable ,Mathematics - Abstract
In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9].
- Published
- 2012
18. Stochastic comparisons of capital allocations with applications
- Author
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Taizhong Hu and Maochao Xu
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Economics and Econometrics ,Mathematical optimization ,Order (exchange) ,media_common.quotation_subject ,Capital (economics) ,Econometrics ,Statistics, Probability and Uncertainty ,Function (engineering) ,Capital allocation line ,Mathematics ,media_common - Abstract
This paper studies capital allocation problems using a general loss function. Stochastic comparisons are conducted for general loss functions in several scenarios: independent and identically distributed risks; independent but non-identically distributed risks; comonotonic risks. Applications in optimal capital allocations and policy limits allocations are discussed as well.
- Published
- 2012
19. Some Inequalities of Linear Combinations of Independent Random Variables. I
- Author
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Taizhong Hu and Maochao Xu
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Statistics and Probability ,Independent and identically distributed random variables ,Exchangeable random variables ,General Mathematics ,media_common.quotation_subject ,peakedness order ,01 natural sciences ,Stochastic ordering ,Combinatorics ,010104 statistics & probability ,Probability theory ,0502 economics and business ,Applied mathematics ,0101 mathematics ,Linear combination ,050205 econometrics ,Mathematics ,media_common ,Pairwise independence ,stochastic order ,Variables ,Likelihood ratio order ,05 social sciences ,Sum of normally distributed random variables ,majorization ,60E15 ,Statistics, Probability and Uncertainty - Abstract
In this paper we provide some sufficient conditions to stochastically compare linear combinations of independent random variables. The main results extend those given in Proschan (1965), Ma (1998), Zhao et al. (2011), and Yu (2011). In particular, we propose a new sufficient condition to compare the peakedness of linear combinations of independent random variables which may have heavy-tailed properties.
- Published
- 2011
20. ORDER STATISTICS FROM HETEROGENOUS NEGATIVE BINOMIAL RANDOM VARIABLES
- Author
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Taizhong Hu and Maochao Xu
- Subjects
Statistics and Probability ,Multivariate statistics ,Mathematical statistics ,Order statistic ,Negative binomial distribution ,Management Science and Operations Research ,Geometric distribution ,Negative multinomial distribution ,Stochastic ordering ,Industrial and Manufacturing Engineering ,Statistics ,Statistics, Probability and Uncertainty ,Mathematics ,L-moment - Abstract
In this article, we study the order statistics from heterogenous negative binomial random variables. Sufficient conditions are provided for comparing the extreme order statistics according to the usual stochastic order. For the special case of geometric distribution, a sufficient condition is established for comparing order statistics in the sense of multivariate stochastic order. Applications in the Poisson–Gamma shock model and redundant systems have been described as well.
- Published
- 2011
21. The tail behavior of the convolutions of Gamma random variables
- Author
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Subhash C. Kochar and Maochao Xu
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,Scale (descriptive set theory) ,Star (graph theory) ,Convolution ,Skewness ,Calculus ,Order (group theory) ,Statistics, Probability and Uncertainty ,Majorization ,Scale parameter ,Random variable ,Mathematics - Abstract
Two sufficient conditions for comparing convolutions of heterogeneous gamma random variables in terms of star order are established. It is further shown that if the scale parameters of heterogeneous gamma random variables are more dispersed in the sense of majorization, then the convolutions are more dispersed according to the right spread order, which generalizes and strengthens the results in Diaconis and Perlman (1987), Kochar and Xu (2010) and Zhao and Balakrishnan (2009).
- Published
- 2011
22. On the right spread order of convolutions of heterogeneous exponential random variables
- Author
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Maochao Xu and Subhash C. Kochar
- Subjects
Statistics and Probability ,Excess wealth order ,Numerical Analysis ,Variables ,Exponential distribution ,NBUE order ,media_common.quotation_subject ,Mathematical analysis ,Lorenz order ,Skewness ,Convolution ,Exponential function ,Distribution (mathematics) ,Sum of normally distributed random variables ,Applied mathematics ,Majorization ,Statistics, Probability and Uncertainty ,Random variable ,media_common ,Mathematics - Abstract
A sufficient condition for comparing convolutions of heterogeneous exponential random variables in terms of right spread order is established. As a consequence, it is shown that a convolution of heterogeneous independent exponential random variables is more skewed than that of homogeneous exponential random variables in the sense of NBUE order. This gives a new insight into the distribution theory of convolutions of independent random variables. A sufficient condition is also derived for comparing such convolutions in terms of Lorenz order.
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- 2010
- Full Text
- View/download PDF
23. ON RESIDUAL LIFETIMES OF k-OUT-OF-n SYSTEMS WITH NONIDENTICAL COMPONENTS
- Author
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Subhash C. Kochar and Maochao Xu
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Hazard (logic) ,Stochastic process ,Proportional hazards model ,Statistics ,Applied mathematics ,Management Science and Operations Research ,Statistics, Probability and Uncertainty ,Residual ,Industrial and Manufacturing Engineering ,Mathematics - Abstract
In this article, mixture representations of survival functions of residual lifetimes of k-out-of-n systems are obtained when the components are independent but not necessarily identically distributed. Then we stochastically compare the residual lifetimes of k-out-of-n systems in one- and two-sample problems. In particular, the results extend some results in Li and Zhao [14], Khaledi and Shaked [13], Sadegh [17], Gurler and Bairamov [7] and Navarro, Balakrishnan, and Samaniego [16]. Applications in the proportional hazard rates model are presented as well.
- Published
- 2009
24. On the range of heterogeneous samples
- Author
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Subhash C. Kochar, Christian Genest, and Maochao Xu
- Subjects
Statistics and Probability ,Numerical Analysis ,021103 operations research ,Order statistic ,0211 other engineering and technologies ,Likelihood ratio ordering ,02 engineering and technology ,Čebyšev’s sum inequality ,Proportional hazards model ,Range ,01 natural sciences ,Order statistics ,Copula (probability theory) ,Monotone regression ,Combinatorics ,Monotone regression dependence ,010104 statistics & probability ,Mean estimation ,Copula ,Exponential distribution ,Calculus ,0101 mathematics ,Statistics, Probability and Uncertainty ,Dispersive ordering ,Mathematics - Abstract
Let R"n be the range of a random sample X"1,...,X"n of exponential random variables with hazard rate @l. Let S"n be the range of another collection Y"1,...,Y"n of mutually independent exponential random variables with hazard rates @l"1,...,@l"n whose average is @l. Finally, let r and s denote the reversed hazard rates of R"n and S"n, respectively. It is shown here that the mapping [email protected]?s(t)/r(t) is increasing on (0,~) and that as a result, R"n=X"("n")-X"("1") is smaller than S"n=Y"("n")-Y"("1") in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X"("n") is seen to be more stochastically increasing in X"("1") than Y"("n") is in Y"("1"). In other words, the pair (X"("1"),X"("n")) is more dependent than the pair (Y"("1"),Y"("n")) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X"1,...,X"n form a random sample from a continuous distribution while Y"1,...,Y"n are mutually independent lifetimes with proportional hazard rates.
- Published
- 2009
25. Comparisons of Parallel Systems According to the Convex Transform Order
- Author
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Subhash Kochar and Maochao Xu
- Subjects
Statistics and Probability ,010104 statistics & probability ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences - Abstract
A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples in the sense of the dispersive order and the right-spread order are established. A sufficient condition is also given for the proportional hazard rate model.
- Published
- 2009
26. Comparisons of Parallel Systems According to the Convex Transform Order
- Author
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Maochao Xu and Subhash C. Kochar
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Exponential distribution ,Component (thermodynamics) ,General Mathematics ,010102 general mathematics ,Order statistic ,Regular polygon ,01 natural sciences ,Exponential function ,010104 statistics & probability ,Probability theory ,Skewness ,Statistics ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples in the sense of the dispersive order and the right-spread order are established. A sufficient condition is also given for the proportional hazard rate model.
- Published
- 2009
27. A new dependence ordering with applications
- Author
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Maochao Xu and Subhash C. Kochar
- Subjects
Frailty model ,Statistics and Probability ,62E10 ,62G30 ,Reverse hazard rate ,Numerical Analysis ,Record value ,Multivariate analysis ,Order statistic ,Copula (linguistics) ,Regression analysis ,Star ordering ,Order statistics ,k-record values ,Distribution function ,Convex transform ordering ,Econometrics ,Statistical physics ,60E15 ,Statistics, Probability and Uncertainty ,Partially ordered set ,Random variable ,62H20 ,Dispersive ordering ,Mathematics - Abstract
In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159–171]. Applications of this partial order to order statistics, k-record values and frailty models are given.
- Published
- 2008
- Full Text
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28. Negative dependence in frailty models
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Maochao Xu and Xiaohu Li
- Subjects
Statistics and Probability ,education.field_of_study ,Applied Mathematics ,Population ,Hazard ratio ,Ratio dependent ,Upper and lower bounds ,Variable (computer science) ,Survival function ,Statistics ,Econometrics ,Statistics, Probability and Uncertainty ,education ,Mathematics - Abstract
This note investigates the negative dependence in frailty models. First, we show that the frailty variable and the overall population variable are negatively likelihood ratio dependent and derive an upper bound for the survival function of the population with higher frailty. Secondly, we prove that the DFR property and the logconvex hazard rate of the baseline variable imply the DLR property of the population variable. Finally, we further prove that the likelihood ratio order, hazard rate order and reversed hazard rate order between two frailty variables imply the likelihood ratio order, reversed hazard rate order, and hazard rate order between the corresponding overall population variables, respectively.
- Published
- 2008
29. Stochastic comparisons of spacings of record values from one or two sample sequences
- Author
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Xiaohu Li, Zhouping Li, Peng Zhao, and Maochao Xu
- Subjects
Statistics and Probability ,Statistics ,Regular polygon ,Order (group theory) ,Two sample ,Statistics, Probability and Uncertainty ,Residual ,Sample sequence ,Mathematics - Abstract
This paper investigates the stochastic comparison of spacings of record values. The likelihood ratio order, the hazard rate order and the mean residual life order between spacings of record values from one sample sequence are developed. Furthermore, the increasing convex order and the mean residual life order between simple spacings of record values from two sample sequences are built as well.
- Published
- 2008
30. STOCHASTIC COMPARISONS OF PARALLEL SYSTEMS WHEN COMPONENTS HAVE PROPORTIONAL HAZARD RATES
- Author
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Maochao Xu and Subhash C. Kochar
- Subjects
Statistics and Probability ,Hazard (logic) ,education.field_of_study ,Stochastic process ,Hazard ratio ,Population ,Mathematical statistics ,Parallel computing ,Management Science and Operations Research ,Industrial and Manufacturing Engineering ,Survival function ,Order (group theory) ,Statistics, Probability and Uncertainty ,education ,Random variable ,Mathematics - Abstract
Let X1, … , Xn be independent random variables with Xi having survival function Fλi, i = 1, … , n, and let Y1, … ,Yn be a random sample with common population survival distribution F, where c = ∑i=1nλi/n. Let Xn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Xn:n − X1:n is larger than Yn:n − Y1:n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively.
- Published
- 2007
31. Excess wealth order and sample spacings
- Author
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Subhash C. Kochar, Xiaohu Li, and Maochao Xu
- Subjects
Statistics and Probability ,Combinatorics ,Order (group theory) ,Sample (statistics) ,Mathematical economics ,Variable (mathematics) ,Mathematics - Abstract
In this note, we further study the properties of excess wealth (or right spread) order and the location independent riskier order. It is proved that if X is less variable than Y according to excess wealth order, then X n : n − X k : n ≤ icx Y n : n − Y k : n for k = 0 , 1 , … , n − 1 , where X 0 : n = Y 0 : n ≡ 0 . Similar results are obtained for location independent riskier order. An application in k -price business auction models is presented as well.
- Published
- 2007
32. Likelihood Ratio Order ofm-Spacings in Multiple-Outlier Models
- Author
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Xiaohu Li, Zhouping Li, Maochao Xu, and Peng Zhao
- Subjects
Statistics and Probability ,Simple (abstract algebra) ,Likelihood-ratio test ,Outlier ,Statistics ,Order statistic ,Order (group theory) ,Sample (statistics) ,Stochastic ordering ,Exponential function ,Mathematics - Abstract
This article deals with the multiple-outlier exponential model. The likelihood ratio order between m-spacings of the combined sample is developed, some results extend the conclusions on simple spacings in Wen et al. (2007).
- Published
- 2007
33. Reversed hazard rate order of equilibrium distributions and a related aging notion
- Author
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Xiaohu Li and Maochao Xu
- Subjects
Statistics and Probability ,Aging property ,Series (mathematics) ,Order (exchange) ,Hazard ratio ,Econometrics ,Nonparametric statistics ,Applied mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Reliability (statistics) ,Mathematics - Abstract
This paper deals with preservation of the reversed hazard rate order between equilibrium random variables under formations of some reliability structures. We further investigate a new aging notion based upon the reversed hazard rate order between a random life and its equilibrium version. A nonparametric method is developed to test the exponentiality against such a strict aging property, some numerical results are presented as well.
- Published
- 2007
34. Likelihood ratio order of m-spacings for two samples
- Author
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Xiaohu Li and Maochao Xu
- Subjects
Statistics and Probability ,education.field_of_study ,Applied Mathematics ,Population ,Sense (electronics) ,Sample spacing ,Combinatorics ,Mean estimation ,Statistics ,Order (group theory) ,Statistics, Probability and Uncertainty ,education ,Random variable ,Mathematics - Abstract
For two non-negative random variables X and Y, it is shown that if X is smaller than Y in the sense of the likelihood ratio order and either X or Y is of decreasing likelihood ratio, then the m-spacing V k : n ( m ) of an X-sample is smaller than W k : n ( m ) , the m-spacing of a Y-sample, in terms of the likelihood ratio order. It is also proved that a similar result for the upshifted likelihood ratio order is valid. Finally, it is proved that the decreasing failure rate in average property of a population X is preserved by sample spacing V k : n ( 1 ) .
- Published
- 2006
35. SOME RESULTS ABOUT MIT ORDER AND IMIT CLASS OF LIFE DISTRIBUTIONS
- Author
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Maochao Xu and Xiaohu Li
- Subjects
Statistics and Probability ,Class (set theory) ,Statistics ,Applied mathematics ,Order (group theory) ,Renewal theory ,Management Science and Operations Research ,Statistics, Probability and Uncertainty ,Residual ,Industrial and Manufacturing Engineering ,Mathematics - Abstract
We investigate some new properties of mean inactivity time (MIT) order and increasing MIT (IMIT) class of life distributions. The preservation property of MIT order under increasing and concave transformations, reversed preservation properties of MIT order, and IMIT class of life distributions under the taking of maximum are developed. Based on the residual life at a random time and the excess lifetime in a renewal process, stochastic comparisons of both IMIT and decreasing mean residual life distributions are conducted as well.
- Published
- 2006
36. Some inequalities of linear combinations of independent random variables: II
- Author
-
Xiaoqing Pan, Maochao Xu, and Taizhong Hu
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Discrete mathematics ,Series (mathematics) ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Distribution (mathematics) ,Monotone polygon ,FOS: Mathematics ,majorization ,usual stochastic order ,log-concavity ,likelihood ratio order ,Linear combination ,Majorization ,Random variable ,Schur-concavity ,Mathematics ,Complement (set theory) - Abstract
Linear combinations of independent random variables have been extensively studied in the literature. However, most of the work is based on some specific distribution assumptions. In this paper, a companion of (J. Appl. Probab. 48 (2011) 1179-1188), we unify the study of linear combinations of independent nonnegative random variables under the general setup by using some monotone transforms. The results are further generalized to the case of independent but not necessarily identically distributed nonnegative random variables. The main results complement and generalize the results in the literature including (In Studies in Econometrics, Time Series, and Multivariate Statistics (1983) 465-489 Academic Press; Sankhy\={a} Ser. A 60 (1998) 171-175; Sankhy\={a} Ser. A 63 (2001) 128-132; J. Statist. Plann. Inference 92 (2001) 1-5; Bernoulli 17 (2011) 1044-1053)., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ429 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
- Published
- 2013
37. On the convolution of heterogeneous Bernoulli random variables
- Author
-
Narayanaswamy Balakrishnan and Maochao Xu
- Subjects
Statistics and Probability ,Mathematical optimization ,62G30 ,General Mathematics ,01 natural sciences ,Convolution ,Bernoulli's principle ,010104 statistics & probability ,62N05 ,0502 economics and business ,Applied mathematics ,Order (group theory) ,62D05 ,0101 mathematics ,Majorization ,likelihood ratio order ,Mathematics ,050205 econometrics ,010102 general mathematics ,05 social sciences ,Bernoulli ,reversed hazard rate order ,heterogeneous variable ,majorization ,Bernoulli process ,Statistics, Probability and Uncertainty ,60E15 ,Random variable - Abstract
In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).
- Published
- 2011
38. On the skewness of order statistics in multiple-outlier models
- Author
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Subhash C. Kochar and Maochao Xu
- Subjects
Independent and identically distributed random variables ,Statistics and Probability ,proportional hazard rate ,62G30 ,Exponential distribution ,decreasing failure rate ,Open problem ,General Mathematics ,skewness ,k-out-of-n system ,star order ,01 natural sciences ,010104 statistics & probability ,62N05 ,Probability theory ,Statistics ,Applied mathematics ,exponential distribution ,62D05 ,0101 mathematics ,Mathematics ,right spread order ,Order statistic ,010102 general mathematics ,Exponential function ,Skewness ,Outlier ,Dispersive order ,Statistics, Probability and Uncertainty ,60E15 - Abstract
Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the general k-out-of-n systems for the case when there are only two types of component in the system. An open problem proposed in Pǎltǎnea (2008) is partially solved.
- Published
- 2011
39. CORRECTION TO 'STOCHASTIC COMPARISONS OF PARALLEL SYSTEMS WHEN COMPONENTS HAVE PROPORTIONAL HAZARD RATES'
- Author
-
Maochao Xu and Subhash Kochar
- Subjects
Statistics and Probability ,Hazard (logic) ,Proportional hazards model ,Stochastic process ,Hazard ratio ,Management Science and Operations Research ,Stochastic ordering ,Industrial and Manufacturing Engineering ,Statistics ,Order (group theory) ,Applied mathematics ,Stochastic optimization ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In our article [3] we have found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order.
- Published
- 2008
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