Large Deviations of the Ergodic Averages: From Hölder Continuity to Continuity Almost Everywhere

For many dynamical systems that are popular in applications, estimates are known for the decay of large deviations of the ergodic averages in the case of Hölder continuous averaging functions. In the present article, we show that these estimates are valid with the same asymptotics in the case of bounded almost everywhere continuous functions. Using this fact, we obtain, in the case of such functions, estimates for the rate of convergence in Birkhoff’s ergodic theorem and for the distribution of the time of return to a subset of the phase space.     

Stability estimates for decompositions of the composition of Gaussian and Poisson distributions

Stability estimates are obtained for decompositions of the composition of Gaussian and Poisson distribution functions which are sharp in the sense of order in the uniform metric; also obtained are two-sided stability estimates of decompositions of these distribution functions in the Levi metric, which are in some sense close to the sharp ones.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 164–186, 1979.     

逐步增加Ⅱ型截尾下Pareto分布的Bayes估计

基于逐步增加的Ⅱ型截尾样本,当Pareto分布的尺度参数已知时,分别在平方损失和LINEX损失下讨论了其形状参数和可靠性指标(失效率和可靠度)的Bayes估计,并用Monte-Carlo方法对估计结果的MSE,进行了模拟比较.结果表明了在LINEX损失下的估计结果更有效.     

Stability of a theorem of uniqueness of a characteristic function that is analytic in the neighborhood of the origin

Quantitative estimates are obtained for the closeness of distribution functions in terms of the closeness of their characteristic functions considered in finite intervals only. A detailed analysis is presented for the characteristic function of the standard normal distribution on a straight line.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 88–94, 1980.     

Bayes type estimates

We study the behavior of statistical estimates of the parameter of distribution when instead of the original family of distribution densities of the random variable considered one uses another family of functions. The limit distributions of Bayes type estimates are found.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 62–68, 1988.     

Sup-norm convergence rates for Lévy density estimation

In this paper, we consider projection estimates for Lévy densities in high-frequency setup. We give a unified treatment for different sets of basis functions and focus on the asymptotic properties of the maximal deviation distribution for these estimates. Our results are based on the idea to reformulate the problems in terms of Gaussian processes of some special type and to further analyze these Gaussian processes. In particular, we construct a sequence of excursion sets, which guarantees the convergence of the deviation distribution to the Gumbel distribution. We show that the exact rates of convergence presented in previous articles on this topic are logarithmic and construct the sequence of accompanying laws, which approximate the deviation distribution with polynomial rate.     

Asymptotic statistical properties of spectral estimates with different tapers for discrete time processes

This paper concerns the problem of estimating the spectral density, spectral distribution and autocovariance functions on non-overlapped and overlapped intervals for a discrete parameter stationary time series with different tapers. Also, we obtain another estimate of spectral density function via different weight functions. Asymptotic statistical properties of these estimates and their limits will be considered.     

Distribution of martingales with bounded square functions

We study the terminate distribution of a martingale whose square function is bounded. We obtain sharp estimates for the exponential and p-moments, as well as for the distribution function itself. The proofs are based on the elaboration of the Burkholder method and on the investigation of certain locally concave functions.     

Asymptotic Estimates of Elementary Probability Distributions

Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions.     

A new approach for parameter estimation of finite Weibull mixture distributions for reliability modeling

The aim of this paper is to model lifetime data for systems that have failure modes by using the finite mixture of Weibull distributions. It involves estimating of the unknown parameters which is an important task in statistics, especially in life testing and reliability analysis. The proposed approach depends on different methods that will be used to develop the estimates such as MLE through the EM algorithm. In addition, Bayesian estimations will be investigated and some other extensions such as Graphic, Non-Linear Median Rank Regression and Monte Carlo simulation methods can be used to model the system under consideration. A numerical application will be used through the proposed approach. This paper also presents a comparison of the fitted probability density functions, reliability functions and hazard functions of the 3-parameter Weibull and Weibull mixture distributions using the proposed approach and other conventional methods which characterize the distribution of failure times for the system components. GOF is used to determine the best distribution for modeling lifetime data, the priority will be for the proposed approach which has more accurate parameter estimates.     

On Robust estimation of a correlation coefficient

Robust estimation of the correlation coefficient of a bivariate normal distribution is considered in the case of a contamination scheme. A number of conventional robust estimates are studied, and some new estimates are proposed. Their properties are examined on finite samples and in asymptotics with the use of Monte-Carlo and the influence functions techniques correspondingly. It is shown that one of the proposed estimates called a median correlation coefficient has high robustness properties. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models. Kazan, Russian, 1995, Part II.     

马氏调制费率的带干扰风险模型

考虑了一类具有马氏调制的带干扰连续时间风险模型,得到了该模型下其条件Gerber-Shiu折现罚金函数所满足的积分方程,Laplace变换及渐近解.在两状态情形下,当索赔额的分布为有理数情况时得到了条件Gerber-Shiu折现罚金函数的具体表达式并给出了数值例子     

E-Bayesian estimation for the Lomax distribution based on type-II censored data

This paper is concerned with using the E-Bayesian method for computing estimates of the unknown parameter and some survival time parameters e.g. reliability and hazard functions of Lomax distribution based on type-II censored data. These estimates are derived based on a conjugate prior for the parameter under the balanced squared error loss function. A comparison between the new method and the corresponding Bayes and maximum likelihood techniques is conducted using the Monte Carlo simulation.     

一种估计Clayton模型中关联参数的方法

在右删失情形下,基于二元风险函数的核估计,我们对Clayton模型中的关联参数给出了一种新的估计方法.新的估计量具有相合性和渐近分布,随机模拟也显示这种估计方法是非常有效的.     

E-Bayesian estimation for the Burr type XII model based on type-2 censoring

This paper is concerned with using the E-Bayesian method [M. Han, Applied Mathematical Modeling (2009) 1915–1922] for computing estimates for the parameter and reliability function of the Burr type XII distribution based on type-2 censored samples. The estimates are obtained based on squared error and LINEX loss functions. A comparison between the new method and the corresponding Bayes and maximum likelihood techniques is made using the Monte Carlo simulation.     

Bayesian estimation for generalized exponential distribution based on progressive type-I interval censoring

In this study, we consider the Bayesian estimation of unknown parameters and reliability function of the generalized exponential distribution based on progressive type-I interval censoring. The Bayesian estimates of parameters and reliability function cannot be obtained as explicit forms by applying squared error loss and Linex loss functions, respectively; thus, we present the Lindley’s approximation to discuss these estimations. Then, the Bayesian estimates are compared with the maximum likelihood estimates by using the Monte Carlo simulations.     

A Pareto model for classical systems

A new Pareto distribution is introduced for pooling knowledge about classical systems. It takes the form of the product of two Pareto probability density functions (pdfs). Various structural properties of this distribution are derived, including its cumulative distribution function (cdf), moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix. Copyright © 2007 John Wiley & Sons, Ltd.     

Distribution function estimates by Wasserstein metric and Bernstein approximation for C-1 functions

The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learningBased on the classical Bernstein approximation, a scheme is presented.To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1functions, which was rarely discussed in the classical approximation theoryFinally, we get a probability estimate by the statistical distance.     

Asymptotic behavior of projection estimates on a circle

Estimates are obtained for the proximity of distribution functions from the proximity of their Fourier coefficient on a circle which are used later to study the asymptotic behavior of projection estimates.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 38–45, 1988.     

Variance computations for functional of absolute risk estimates

We present a simple influence function based approach for computing the variances of estimates of absolute risk and functions of absolute risk. We apply this approach to criteria that assess the impact of changes in the risk factor distribution on absolute risk for an individual and at the population level. As an illustration we use an absolute risk prediction model for breast cancer that includes modifiable risk factors in addition to standard breast cancer risk factors. Influence function based variance estimates for absolute risk and the criteria are compared to bootstrap variance estimates.