ESTIMATION OF THE VARIANCE FOR STRONGLY MIXING SEQUENCES

Let {X\-n,n≥1} be a stationary strongly mixing random sequence satisfying E X\-1=μ, E X\+2\-1<∞ and (Var S\-n)/n→σ\+2 as n→∞ . In this paper a class of estimators of Var S\-n is studied. The weak consistency and asymptotic normality as well as the central limit theorem are presented.

On p-variation of bifractional Brownian motion

In this paper we study p-variation of bifractional Brownian motion. As an applica-tion, we introduce a class of estimators of the parameters of a bifractional Brownian motion andprove that both of them are strongly consistent; as another application, we investigate fractalnature related to the box dimension of the graph of bifractional Brownian motion.

POWER VARIATION OF SUBFRACTIONAL BROWNIAN MOTION AND APPLICATION

In this paper, we consider the power variation of subfractional Brownian mo- tion. As an application, we introduce a class of estimators for the index of a subfractional Brownian motion and show that they are strongly consistent.

Sieve MLE for Generalized Partial Linear Models with Type Ⅱ Interval-censored Data

This article concerded with a semiparametric generalized partial linear model （GPLM） with the type Ⅱ censored data. A sieve maximum likelihood estimator （MLE） is proposed to estimate the parameter component, allowing exploration of the nonlinear relationship between a certain covariate and the response function. Asymptotic properties of the proposed sieve MLEs are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. Moreover, the estimators of the unknown parameters are asymptotically normal and efficient, and the estimator of the nonparametric function has an optimal convergence rate.

EXACT MAXIMUM LIKELIHOOD ESTIMATOR FOR DRIFT FRACTIONAL BROWNIAN MOTION AT DISCRETE OBSERVATION

This paper deals with the problems of consistency and strong consistency of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. Both the central limit theorem and the Berry-Ess′een bounds for these estimators are obtained by using the Stein’s method via Malliavin calculus.

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

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.

ASYMPTOTIC PROPERTIES OF ESTIMATORS IN PARTIALLY LINEAR SINGLE-INDEX MODEL FOR LONGITUDINAL DATA

In this article, a partially linear single-index model /or longitudinal data is investigated. The generalized penalized spline least squares estimates of the unknown parameters are suggested. All parameters can be estimated simultaneously by the proposed method while the feature of longitudinal data is considered. The existence, strong consistency and asymptotic normality of the estimators are proved under suitable conditions. A simulation study is conducted to investigate the finite sample performance of the proposed method. Our approach can also be used to study the pure single-index model for longitudinal data.

Rate of strong consistency of the maximum quasi-likelihood estimator in quasi-likelihood nonlinear models

Quasi-likelihood nonlinear models （QLNM） include generalized linear models as a special case. Under some regularity conditions, the rate of the strong consistency of the maximum quasi-likelihood estimation （MQLE） is obtained in QLNM. In an important case, this rate is O（n-^1/2（loglogn）^1/2）, which is just the rate of LIL of partial sums for i.i.d variables, and thus cannot be improved anymore.

PARAMETER ESTIMATION IN LINEAR REGRESSION MODELS FOR LONGITUDINAL CONTAMINATED DATA

The parameter estimation and the coefficient of contamination for the regression models with repeated measures are studied when its response variables are contaminated by another random variable sequence.Under the suitable conditions it is proved that the estimators which are established in the paper are strongly consistent estimators.

NON-PARAMETRIC ESTIMATION IN CONTAMINATED LINEAR MODEL

In this paper, the following contaminated linear model is considered:y i=(1-ε)x τ iβ+z i, 1≤i≤n,where r.v.'s ｛ y i ｝ are contaminated with errors ｛ z i ｝. To assume that the errors have the finite moment of order 2 only. The non parametric estimation of contaminated coefficient ε and regression parameter β are established, and the strong consistency and convergence rate almost surely of the estimators are obtained. A simulated example is also given to show the visual performance of the estimations.

NON-PARAMETRIC LEAST SQUARE ESTIMATION OF DISTRIBUTION FUNCTION

By using the non parametric least square method, the strong consistent estimations of distribution function and failure function are established,where the distribution function F(x) after logist transformation is assumed to be approximated by a polynomial.The performance of simulation shows that the estimations are highly satisfactory.

SOME LARGE SAMPLE PROPERTIES OF AN ESTIMATOR OF THE HAZARD FUNCTION FROM RANDOMLY CENSORED DATA

In this paper, A nonparametric hazard estimator is introduced. Weak convergence and strong uniformly consistency of the proposed estimator lambda(n)(t) are investigated on a bounded interval, respectively. An asymptotic representation of lambda(n)(t) is also given, and the asymptotic representation is used to prove asymptotic normality of the hazard estimator.

PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS

This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters.First,we prove the existence and uniqueness of these equations under non-Lipschitz conditions.Second,we construct maximum likelihood estimators of these parameters and then discuss their strong consistency.Third,a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered.Finally,we estimate the errors between solutions of these equations and that of their numerical equations.

SEMIPARAMETRIC REGRESSION MODELS WITH LOCALLY GENERALIZED GAUSSIAN ERROR'S STRUCTURE

This paper proposes parametric component and nonparametric component estimators in a semiparametric regression models based on least squares and weight function's method, their strong consistency and rib mean consistency are obtained under a locally generallied Gaussinan error's structure. Finally, the author showes that the usual weight functions based on nearest neighbor method satisfy the deigned assumptions imposed.

Nonparametric estimation for contamination distribution

In the paper, for the contamination distribution model F（x） = （1-α）F1（x）＋αF2（x）, the estimates of α and F1 （x） are studied using two different ways when F2 （x） is known and the strong consistency of the two estimates is proved. At the same time the consistency rate of estimate α is also given.

Strong Consistency of M Estimator in Linear Model for φ-mixing Samples

The strong consistency of M estimator of regression parameter in linear model for φ-mixing samples is discussed by using the classic Rosenthal type inequality. We get the strong consistency of M estimator under lower moment condition, which generalizes and improves the corresponding ones for independent sequences.

A KIND OF URN MODEL FOR ADAPTIVE SEQUENTIAL DESIGN

This paper proposes a new kind of generalized Friendman's urn model,which with adaptive nonhomogeneous generating matrix.This model may be applied in sequential medical experiment.In this model some limit theorems (strong consistency and asymptot- ical normality) have been obtained.

Estimation of Distribution Function Based on Presmoothed Relative-Risk Function

In this article, the lifetime data subjecting to right random censoring is considered. Nonparametric estimation of the distribution function based on the conception of presmoothed estimation of relative-risk function and the properties of the estimator by using methods of numerical modeling are discussed. In the model under consideration, the estimates were compared using numerical methods to determine which of the estimates is actually better.

Local Polynomial Estimation of Distribution Functions

Under the condition that the total distribution function is continuous and bounded on （ -∞,∞ ）, we constructed estimations for distribution and hazard functions with local polynomial method, and obtained the rate of strong convergence of the estimations.

Strong Consistency of the Spline-Estimation of Probabilities Density in Uniform Metric

In the present paper as estimation of an unknown probability density of the spline-estimation is constructed, necessity and sufficiency conditions of strong consistency of the spline-estimation are given.