Consistency and asymptotic normality of the maximum quasi-likelihood estimator in quasi-likelihood nonlinear models with random regressors

This paper proposes some regularity conditions, which result in the existence, strong consistency and asymptotic normality of maximum quasi-likelihood estimator （MQLE） in quasi-likelihood nonlinear models （QLNM） with random regressors. The asymptotic results of generalized linear models （GLM） with random regressors are generalized to QLNM with random regressors.     

Asymptotic normality and strong consistency of maximum quasi-likelihood estimates in generalized linear models

In a generalized linear model with q x 1 responses, the bounded and fixed (or adaptive) p × q regressors Zi and the general link function, under the most general assumption on the minimum eigenvalue of ZiZ'i,the moment condition on responses as weak as possible and the other mild regular conditions, we prove that the maximum quasi-likelihood estimates for the regression parameter vector are asymptotically normal and strongly consistent.     

Strong consistency of maximum quasi-likelihood estimates in generalized linear models

In a generalized linear model with q×1 responses, bounded and fixed p×q regressors zi and general link function, under the most general assumption on the minimum eigenvalue of ∑in=1 ZiZi', the moment condition on responses as weak as possible and other mild regular conditions, we prove that with probability one, the quasi-likelihood equation has a solution βn for all large sample size n, which converges to the true regression parameter β0. This result is an essential improvement over the relevant results in literature.     

Rate of strong consistency of quasi maximum likelihood estimate in generalized linear models

Under the assumption that in the generalized linear model (GLM) the expectation of the response variable has a correct specification and some other smooth conditions, it is shown that with probability one the quasi-likelihood equation for the GLM has a solution when the sample size n is sufficiently large. The rate of this solution tending to the true value is determined. In an important special case, this rate is the same as specified in the LIL for iid partial sums and thus cannot be improved anymore.     

拟似然非线性模型中最大拟似然估计的强相合性

This paper proposes some regularity conditions. On the basis of the proposed regularity conditions, we show the strong consistency of maximum quasi-likelihood estimation （MQLE） in quasi-likelihood nonlinear models （QLNM）. Our results may be regarded as a further generalization of the relevant results in Ref. [4].     

拟似然非线性模型中MQLE的弱相合性的充分条件

拟似然非线性模型包括广义线性模型作为一个特殊情形.给出了拟似然非线性模型中极大拟似然估计的弱相合性的一些充分条件,其中矩的条件要弱于文献中极大拟似然估计的强相合性的条件.     

拟似然非线性模型的某些渐近推断:几何方法

Abstract. A modified Bates and Watts geometric framework is proposed for quasi-likelihoodnonlinear models in Euclidean inner product space. Based on the modified geometric framework,some asymptotic inference in terms of curvatures for quasi-likelihood nonlinear models is stud-ied. Several previous results for nonlinear regression models and exponential family nonlinearmodels etc. are extended to quasi-likelihood nonlinear models.     

The rate of strong uniform consistency for the multivariate product-limit estimator

The general asymptotic order of magnitude is determined for the maximal deviation of the multivariate product-limit estimate from the estimated survival function on R^k. This order depends on the joint behavior of the censoring and censored distributions in a well-defined way. Corresponding to specific joint behaviors, several lim sup results are deduced generalizing everything that is known in the univariate case. The results are also extended for the variable censoring model.     

Asymptotic properties of maximum likelihood estimator for two-step logit models

Two-step logit models are extensions of the ordinary logistic regression model, which are designed for complex ordinal outcomes commonly seen in practice. In this paper, we establish some asymptotic properties of the maximum likelihood estimator (MLE) of the regression parameter vector under some mild conditions, which include existence of the MLE, convergence rate and asymptotic normality of the MLE. We relax the boundedness condition of the regressors required in most existing theoretical results, and all conditions are easy to verify.     

The strong consistency of M-estimators in linear models

The strong consistency of M-estimators in linear models is considered. Under some conditions on the ratios of maximum and minimum eigenvalues of the information matrices the desired result is established.     

自适应设计广义线性模型极大拟似然估计的渐近性

在对Fisher信息矩阵的最小特征根最一般的假定,响应变量的矩条件尽可能弱和其它正则条件下,证明了自适应设计广义线性模型中极大拟似然估计的强相合性与渐近正态性,同时给出了强收敛速度.     

Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator

This paper deals with a general class of observation-driven time series models with a special focus on time series of counts. We provide conditions under which there exist strict-sense stationary and ergodic versions of such processes. The consistency of the maximum likelihood estimators is then derived for well-specified and misspecified models.     

Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process

We consider an estimation problem with observations from a Gaussian process. The problem arises from a stochastic process modeling of computer experiments proposed recently by Sacks, Schiller, and Welch. By establishing various representations and approximations to the corresponding log-likelihood function, we show that the maximum likelihood estimator of the identifiable parameter θσ² is strongly consistent and converges weakly (when normalized by √n) to a normal random variable, whose variance does not depend on the selection of sample points. Some extensions to regression models are also obtained.     

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.     

Strong consistency of M-estimates in linear models

The strong consistency of M-estimates of the regression coefficients in a linear model under some mild conditions is established, which is an essential improvement over the relevant results in the literature on the moment condition. Especially, in some important circumstances, onlyE|ψ(e_k)|^q for some q > 1 is needed, where ψ{e_k} is some score function of random error.     

指数族非线性模型最大似然估计的相合性和渐近正态性

本文我们提出了一些正则条件, 这些条件减弱了Zhu and Wei (1997)文中的条件. 基于所提的正则条件, 我们证明了指数族非线性模型参数最大似然估计的相合性和渐近正态性. 我们的结果可被认为是Zhu and Wei (1997)工作的进一步改进.     

Strong consistency of MLE for finite uniform mixtures when the scale parameters are exponentially small

We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from below by exp(−n ^d ), 0<d<1, wheren is the sample size.     

Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters

This paper establishes several almost sure asymptotic properties of general autoregressive processes. By making use of these properties, we obtain a proof of the strong consistency of the least-squares estimates of the parameters of the process without any assumption on the roots of the characteristic polynomial.     

Convergence systems and strong consistency of least squares estimates in regression models

A recent theorem of T. L. Hai, H. Robbins, and C. Z. Wei (J. Multivariate Anal.9 (1979), 343–362) is extended to a more general form which unifies previous results in the literature on the strong consistency of least squares estimates in multiple regression models with nonrandom regressors. In particular the issue of strong consistency of the least squares estimate in the Gauss-Markov model, in the i.i.d. model with infinite second moment, and in general time series models is examined. In this connection, some basic properties of convergence systems are also obtained and are applied to the strong consistency problem.     

On a robust and efficient maximum depth estimator

The best breakdown point robustness is one of the most outstanding features of the univariate median. For this robustness property, the median, however, has to pay the price of a low efficiency at normal and other light-tailed models. Affine equivariant multivariate analogues of the univariate median with high breakdown points were constructed in the past two decades. For the high breakdown robustness, most of them also have to sacrifice their efficiency at normal and other models, nevertheless. The affine equivariant maximum depth estimator proposed and studied in this paper turns out to be an exception. Like the univariate median, it also possesses a highest breakdown point among all its multivariate competitors. Unlike the univariate median, it is also highly efficient relative to the sample mean at normal and various other distributions, overcoming the vital low-efficiency shortcoming of the univariate and other multivariate generalized medians. The paper also studies the asymptotics of the estimator and establishes its limit distribution without symmetry and other strong assumptions that are typically imposed on the underlying distribution. This work was supported by Natural Science Foundation of USA (Grant Nos. DMS-0071976, DMS-0234078) and by the Southwestern University of Finance and Economics Third Period Construction Item Funds of the 211 Project (Grant No. 211D3T06)