Estimation for the semipareto processes

This paper proposes different estimators for the parameters of SemiPareto and Pareto autoregressive minification processes The asymptotic properties of the estimators are established by showing that the SemiPareto process is α-mixing. Asymptotic variances of different moment and maximum likelihood estimators are compared.     

Robust kernel estimators for additive models with dependent observations

Robust nonparametric estimators for additive regression or autoregression models under an α-mixing condition are proposed. They are based on local M-estimators or local medians with kernel weights, and their asymptotic behaviour is studied. Moreover, diese local M-estimators achieve the same univariate rate of convergence as their linear relatives.     

Nonparametric Estimation of the Geometric Vitality Function

Recently, Nair and Rajesh (2000 Nair , K. R. M. , Rajesh , G. ( 2000 ). Geometric vitality function and its application to reliability . IAPQR Tran. 25 ( 1 ): 1 – 8 . [Google Scholar]) proposed a measure to describe the failure pattern of components/devices in terms of the geometric mean of the residual life. This measure find applications in modeling life time data. In the present work we provide a nonparametric kernel-type estimator for the geometric vitality function, both in the case of complete and censored samples. The properties of the estimator, under certain regularity conditions, are studied. The performance of the estimator is compared with the empirical estimator using a real data set and simulation studies are carried out using the Monte Carlo method.     

On Complete Convergence for Nonstationary ϕ-Mixing Random Variables

In this article, we study the complete convergence for non-stationary ?-mixing random variables, especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for ?-mixing random variables. Our result generalizes the corresponding one of Shao (1988 Shao, Q.M. (1988). A moment inequality and its applications. Acta Mathematica Sinica 31(6):736–747. [Google Scholar]) and improves the corresponding one of Peligrad (1985a Peligrad, M. (1985a). Convergence rates of the strong law for stationary mixing sequences. Z. Wahrsch. verw. Gebiete. 70:307–314.[Crossref], [Web of Science ®] , [Google Scholar]) and Wang (1987 Wang, D.C. (1987). The complete convergence of the sums for the stationary sequence of ?-mixing random variables. J. Syst. Sci. Mathemat. Sci. 7(4):352–359. [Google Scholar]).     

Invariance principles in probability for stable processes generated by a class of dependent sequences

We prove an invariance principle in probability for φ-mixing sequences of random variables taking values in a separable Banach space. The conditions on the dependence structure required for this approximation result are known to be necessary and sufficient for weak convergence of the partial-sum process to the appropriate limiting process.     

On the rate of convergence of the Robbins–Monro's algorithm in a linear stochastic ill-posed problem with α-mixing data

In this paper we consider a recursive method of Robbins–Monro type to estimate the solution of the linear problem Ax = u, in which the second member is measured with α-mixing errors. We also show the almost complete convergence (a.co) of this algorithm specifying its convergence rate.     

An invariance principle for triangular arrays of dependent variables with application to autocovariance estimation

The invariance principle for triangular arrays of dependent variables is studied. We use the concept of mixingale, proposed by McLeish (1975). Uniform bounds are imposed on the growth of conditional expectations with respect to distant predecessors. The theorem is applied to invariance principles for autocovariance estimates based on triangular arrays of time-series data for weak mixing sequences and linear processes. Such results are required for bootstrap applications.