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On the Second Borel-Cantelli Lemma for α-mixing Sequences of Events

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Chinese Quarterly Journal of Mathematics 2009年第4期24卷 551-560页

作者： YUAN De-mei College of Science Chongqing Technology and Business University Chongqing 400067 China

In this paper, we give some conditions on diverging rate of series of the probabilities and converging rate of series of the α-mixing coefficients for sequences of events, under which the conclusion of the Second Borel-Cantelli Lemma holds. As corollaries, some moment conditions are obtained, under which the strong law of large numbers holds for sequences of identically distributed random variables.

关键词： Borel-Cantelli lemma strong mixing α-mixing strong law of large numbers

Complete convergence for a-mixing sequence

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Chinese Science Bulletin 1997年第13期42卷 1068-1073页

作者： XU BingNormal College, Ningbo University, Ningbo 315020, China Normal College Ningbo University Ningbo 315020 China

1 Main resultsSINCE the concept of complete convergence was raised by Hsu and Robbins （1947）, it has at- tracted much attention. The general results were proved by Baum and Katz （1965）: if{Xn,n≥1} is a sequence o... 详细信息

关键词： α-mixing complete convergence maximum inequalities.

Identification of Linear Systems Using Binary Sensors with Random Thresholds

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Journal of Systems Science & Complexity 2024年第3期37卷 907-923页

作者： HUANG Zhiyong SONG Qijiang School of Mathematics Renmin University of ChinaBeijing 100872China School of Mathematics Sino-French InstituteRenmin University of ChinaBeijing 100872China

In this paper,the problem of identifying autoregressive-moving-average systems under random threshold binary-valued output measurements is *** the help of stochastic approximation algorithms with expanding truncations,the authors give the recursive estimates for the parameters of both the linear system and the binary *** reasonable conditions,all constructed estimates are proved to be convergent to the true values with probability one,and the convergence rates are also established.A simulation example is provided to justify the theoretical results.

关键词： α-mixing binary sensor quantized system stochastic approximation stochastic thresh-old strongly consistent system identification

Uncertainty Comparison Between Value-at-Risk and Expected Shortfall

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Communications in Mathematical Research 2024年第1期40卷 102-124页

作者： Qing Liu Weimin Liu Liang Peng Gengsheng Qin School of Statistics Jjiangxi University of Finance and EconomicsNanchangJiangxi 330013China Department of Mathematics and Statistics Georgia State UniversityAtlantaGA 30303USA Maurice R.Greenberg School of Risk Science Georgia State UniversityAtlantaGA 20303USA

Value-at-Risk(VaR)and expected shortfall(ES)are two key risk measures in financial risk *** these two measures has been a hot debate,and most discussions focus on risk measure *** paper uses independent data and autoregressive models with normal or t-distribution to examine the effect of the heavy tail and dependence on comparing the nonparametric inference uncertainty of these two risk *** and numerical analyses suggest that VaR at 99%level is better than ES at 97.5%level for distributions with heavier tails.

关键词： α-mixing asymptotic variance expected shortfall Value-at-Risk

NONPARAMETRIC APPROACH TO IDENTIFYING NARX SYSTEMS

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Journal of Systems Science & Complexity 2010年第1期23卷 3-21页

作者： Qijiang SONG·Han-Pu CHEN Key Laboratory of Systems and Control,Institute of Systems Science,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China Key Laboratory of Systems and Control Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

This paper considers identification of the nonlinear autoregression with exogenous inputs(NARX system).The growth rate of the nonlinear function is required be not faster than linear withslope less than *** value of f(·) at any fixed point is recursively estimated by the stochasticapproximation (SA) algorithm with the help of kernel *** consistency of the estimatesis established under reasonable conditions,which,in particular,imply stability of the *** simulation is consistent with the theoretical analysis.

关键词： α-mixing geometrically ergodic Markov chains NARX nonparametric recursive estimate stochastic approximation strongly consistent.

Identification of Errors-in-Variables Systems with General Nonlinear Output Observations and with ARMA Observation Noises

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Journal of Systems Science & Complexity 2020年第1期33卷 1-14页

作者： SONG Qijiang HUANG Zhiyong School of Mathematics Renmin University of ChinaBeijing 100872China

This paper concerns the identification problem of scalar errors-in-variables(EIV)systems with general nonlinear output observations and ARMA observation *** independent and identically distributed(i.i.d.)Gaussian inputs with unknown variance,recursive algorithms for estimating the parameters of the EIV systems are *** general nonlinear observations,conditions on the system are imposed to guarantee the almost sure convergence of the estimates.A simulation example is included to justify the theoretical results.

关键词： ARMA noise α-mixing binary sensor errors-in-variables nonlinear observation recursive estimate stochastic approximation(SA) strongly consistent system identification

KERNEL ESTIMATION OF HIGHER DERIVATIVES OF DENSITY AND HAZARD RATE FUNCTION FOR TRUNCATED AND CENSORED DEPENDENT DATA

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Acta Mathematica Scientia 2003年第4期23卷 477-486页

作者：陈清平戴永隆 Department of Statistics Wuhan University Wuhan 430072 China 2. Wuhan University of Science and Technology Wuhan 430081 China Department of Statistics Zhongshan University Guangzhou 510275 China

Based on left truncated and right censored dependent data, the estimators of higher derivatives of density function and hazard rate function are given by kernel smoothing method. When observed data exhibit α-mixing dependence, local properties including strong consistency and law of iterated logarithm are presented. Moreover, when the mode estimator is defined as the random variable that maximizes the kernel density estimator, the asymptotic normality of the mode estimator is established.

关键词： Truncated and censored data α-mixing strong consistency law of iterated logarithm mode

Conditional Quantile Estimation with Truncated,Censored and Dependent Data

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Chinese Annals of Mathematics,Series B 2015年第6期36卷 969-990页

作者： Hanying LIANG Deli LI Tianxuan MIAO Department of Mathematics Tongji University Department of Mathematical Sciences Lakehead University

This paper deals with the conditional quantile estimation based on left-truncated and right-censored *** that the observations with multivariate covariates form a stationary α-mixing sequence,the authors derive the strong convergence with rate,strong representation as well as asymptotic normality of the conditional quantile ***,a Berry-Esseen-type bound for the estimator is *** addition,the finite sample behavior of the estimator is investigated via simulations.

关键词： Berry-Esseen-type bound Conditional quantile estimator Strong rep-resentation Truncated and censored data α-mixing

Confidence Intervals of Variance Functions in Generalized Linear Model

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Acta Mathematicae Applicatae Sinica 2006年第3期22卷 353-368页

作者： Yong Zhou Dao-ji Li Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China

In this paper we introduce an appealing nonparametric method for estimating variance and conditional variance functions in generalized linear models （GLMs）, when designs are fixed points and random variables respectively, Bias-corrected confidence bands are proposed for the （conditional） variance by local linear smoothers. Nonparametric techniques are developed in deriving the bias-corrected confidence intervals of the （conditional） variance. The asymptotic distribution of the proposed estimator is established and show that the bias-corrected confidence bands asymptotically have the correct coverage properties. A small simulation is performed when unknown regression parameter is estimated by nonparametric quasi-likelihood. The results are also applicable to nonparamctric autoregressive times series model with heteroscedastic conditional variance.

关键词： Nonlinear time series model variance function conditional heteroscedastie variance generalized linear model local polynomial fitting α-mixing