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.     

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.     

Optimization algorithm with probabilistic estimation

In this paper, we present a stochastic optimization algorithm based on the idea of the gradient method which incorporates a new adaptive-precision technique. Because of this new technique, unlike recent methods, the proposed algorithm adaptively selects the precision without any need for prior knowledge on the speed of convergence of the generated sequence. With this new technique, the algorithm can avoid increasing the estimation precision unnecessarily, yet it retains its favorable convergence properties. In fact, it tries to maintain a nice balance between the requirements for computational accuracy and those for computational expediency. Furthermore, we present two types of convergence results delineating under what assumptions what kinds of convergence can be obtained for the proposed algorithm.The work reported here was supported in part by NSF Grant No. ECS-85-06249 and USAF Grant No. AFOSR-89-0518. The authors wish to thank the anonymous reviewers whose careful reading and criticism have helped them improve the paper considerably.     

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.     

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.     

Quasi-likelihood or extended quasi-likelihood? An information-geometric approach

When the variance is a known function of the mean, as in quasi-likelihood applications, the sample variance also contains information about the mean and extensions of quasi-likelihood functions have been suggested that incorporate this additional information. In order to be sure these extensions are an improvement, further assumptions are made typically on the higher moments of the data so that there is a trade-off between the greater robustness of the quasi-likelihood estimates and the potentially improved estimates based on the extended quasi-likelihood functions. Improvement is often measured by relative efficiency but more insight can be gained by considering optimality of estimating functions, information loss, and sufficiency. All these measures can be described using the dual geometries of the quasi- and extended quasi-likelihood estimators. For a substantial range of models, the extended estimates offer little improvement when the coefficient of variation is small.     

On stochastic estimation

We consider a local random searching method to approximate a root of a specified equation. If such roots, which can be regarded as estimators for the Euclidean parameter of a statistical experiment, have some asymptotic optimality properties, the local random searching method leads to asymptotically optimal estimators in such cases. Application to simple first order autoregressive processes and some simulation results for such models are also included.     

Parameter estimation in general state-space models using particle methods

Particle filtering techniques are a set of powerful and versatile simulation-based methods to perform optimal state estimation in nonlinear non-Gaussian state-space models. If the model includes fixed parameters, a standard technique to perform parameter estimation consists of extending the state with the parameter to transform the problem into an optimal filtering problem. However, this approach requires the use of special particle filtering techniques which suffer from several drawbacks. We consider here an alternative approach combining particle filtering and gradient algorithms to perform batch and recursive maximum likelihood parameter estimation. An original particle method is presented to implement these approaches and their efficiency is assessed through simulation.     

Approximate maximum likelihood estimation in linear regression

The application of the ML method in linear regression requires a parametric form for the error density. When this is not available, the density may be parameterized by its cumulants ( _i ) and the ML then applied. Results are obtained when the standardized cumulants ( _i ) satisfy _i = _i+2/ ₂ ^(i+2)/2 =O(v ⁱ ) asv 0 fori>0.Research financed in part by the Research Center of the Athens University of Economics and Business.     

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.     

Comparison of certain value-at-risk estimation methods for the two-parameter Weibull loss distribution

The Weibull distribution is one of the most important distributions that is utilized as a probability model for loss amounts in connection with actuarial and financial risk management problems. This paper considers the Weibull distribution and its quantiles in the context of estimation of a risk measure called Value-at-Risk (VaR). VaR is simply the maximum loss in a specified period with a pre-assigned probability level. We attempt to present certain estimation methods for VaR as a quantile of a distribution and compare these methods with respect to their deficiency (Def) values. Along this line, the results of some Monte Carlo simulations, that we have conducted for detailed investigations on the efficiency of the estimators as compared to MLE, are provided.     

Non-linear model parameter estimation–estimating a feasible parameter set with respect to model use

This article deals with non-linear model parameter estimation from experimental data. As for non-linear models a rigorous identifiability analysis is difficult to perform, parameter estimation is performed in such a way that uncertainty in the estimated parameter values is represented by the range of model use results when the model is used for a certain purpose. Using this approach, the article presents a simulation study where the objective is to discover whether the estimation of model parameters can be improved, so that a small enough range of model use results is obtained. The results of the study indicate that from plant measurements available for the estimation of model parameters, it is possible to extract data that are important for the estimation of model parameters relative to a certain model use. If these data are improved by a proper measurement campaign (e.g. proper choice of measured variables, better accuracy, higher measurement frequency) it is to be expected that a valid model for a certain model use will be obtained. The simulation study is performed for an activated sludge model from wastewater treatment, while the estimation of model parameters is done by Monte Carlo simulation.     

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.     

Bayesian total loss estimation using shared random effects

The pricing of insurance policies requires estimates of the total loss. The traditional compound model imposes an independence assumption on the number of claims and their individual sizes. Bivariate models, which model both variables jointly, eliminate this assumption. A regression approach allows policy holder characteristics and product features to be included in the model. This article presents a bivariate model that uses joint random effects across both response variables to induce dependence effects. Bayesian posterior estimation is done using Markov Chain Monte Carlo (MCMC) methods. A real data example demonstrates that our proposed model exhibits better fitting and forecasting capabilities than existing models.     

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.     

Asymptotic properties of a particular nonlinear regression quantile estimation

In this paper we shall be concerned with the asymptotic properties of the regression quantile estimation in the nonlinear regression time series models. For these, first we prove the strong consistency and derive the asymptotic normality of the regression quantile estimators for a particular sinusoidal regression model with a simple harmonic component. Next, we extend the results to more complicated sinusoidal models of several harmonic components.     

Monte carlo estimation of the sampling distribution of nonlinear model parameter estimators

The sampling distribution of parameter estimators can be summarized by moments, fractiles or quantiles. For nonlinear models, these quantities are often approximated by power series, approximated by transformed systems, or estimated by Monte Carlo sampling. A control variate approach based on a linear approximation of the nonlinear model is introduced here to reduce the Monte Carlo sampling necessary to achieve a given accuracy. The particular linear approximation chosen has several advantages: its moments and other properties are known, it is easy to implement, and there is a correspondence to asymptotic results that permits assessment of control variate effectiveness prior to sampling via measures of nonlinearity. Empirical results for several nonlinear problems are presented.This research was supported in part by the Office of Naval Research under Contract N00014-79-C-0832.     

Efficient MCMC estimation of some elliptical copula regression models through scale mixtures of normals

This paper proposes an efficient estimation method for some elliptical copula regression models by expressing both copula density and marginal density functions as scale mixtures of normals (SMN). Implementing these models using the SMN is novel and allows efficient estimation via Bayesian methods. An innovative algorithm for the case of complex semicontinuous margins is also presented. We utilize the facts that copulas are invariant to the location and scale of the margins; all elliptical distributions have the same correlation structure; and some densities can be represented by the SMN. Two simulation studies, one on continuous margins and the other on semicontinuous margins, highlight the favorable performance of the proposed methods. Two empirical studies, one on the US excess returns and one on the Thai wage earnings, further illustrate the applicability of the proposals.     

Sampling designs for regression coefficient estimation with correlated errors

The problem of estimating regression coefficients from observations at a finite number of properly designed sampling points is considered when the error process has correlated values and no quadratic mean derivative. Sacks and Ylvisaker (1966,Ann. Math. Statist.,39, 66–89) found an asymptotically optimal design for the best linear unbiased estimator (BLUE). Here, the goal is to find an asymptotically optimal design for a simpler estimator. This is achieved by properly adjusting the median sampling design and the simpler estimator introduced by Schoenfelder (1978, Institute of Statistics Mimeo Series No. 1201, University of North Carolina, Chapel Hill). Examples with stationary (Gauss-Markov) and nonstationary (Wiener) error processes and with linear and nonlinear regression functions are considered both analytically and numerically.Research supported by the Air Force Office of Scientific Research Contract No. 91-0030.