LAN, LAM and convergence of Iterative ML estimates; optimal design in Gaussian one-way mixed model

For a one-way mixed Gaussian ANOVA model we prove local asymptotic normality and local asymptotic minimaxity of maximum likelihood estimates (MLE) and of its certain iterative approximations. A geometric rate of convergence in probability is proved for these iterative estimates to MLE. Asymptotically optimal designs for large samples are studied.     

Parameter identification for the discretely observed geometric fractional Brownian motion

This paper deals with the problem of estimating all the unknown parameters of geometric fractional Brownian processes from discrete observations. The estimation procedure is built upon the marriage of the quadratic variation and the maximum likelihood approach. The asymptotic properties of the estimators are provided. Moveover, we compare our derived method with the approach proposed by Misiran et al. [Fractional Black-Scholes models: complete MLE with application to fractional option pricing. In International conference on optimization and control; Guiyang, China; 2010. p. 573–586.], namely the complete maximum likelihood estimation. Simulation studies confirm theoretical findings and illustrate that our methodology is efficient and reliable. To show how to apply our approach in realistic contexts, an empirical study of Chinese financial market is also presented.     

Estimation in the presence of incidental parameters

This paper considers the estimation of “structural” parameters when the number of unknown parameters increases with the sample size. Neyman and Scott (1948) had demonstrated that maximum likelihood estimators (MLE) of structural parameters may be inconsistent in this case. Patefield (1977) further observed that the asymptotic covariance matrix of the MLE is not equal to the inverse of the information matrix. In this paper we establish asymptotic properties of estimators (which include in particular the MLE) obtained via the usual likelihood approach when the incidental parameters are first replaced by their estimates (which are allowed to depend on the structural parameters). Conditions for consistency and asymptotic normality together with a proper formula for the asymptotic covariance matrix are given. The results are illustrated and applied to the problem of estimating linear functional relationships, and mild conditions on the incidental parameters for the MLE (or an adjusted MLE) to be consistent and asymptotically normal are obtained. These conditions are weaker than those imposed by previous authors.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Estimation of system reliability for exponential distributions based on L ranked set sampling

Abstract

In the case where strength and stress both follow exponential distributions, this paper considers the maximum likelihood estimator (MLE) of the system reliability based on L ranked set sampling (LRSS). The proposed MLE is shown to have existence, uniqueness and asymptotic normality, and its asymptotic variance is obtained by the Fisher information matrix of LRSS. The values of asymptotic relative efficiencies show that the proposed MLE is always more efficient than the MLE using simple random sampling (SRS). However, the MLE using LRSS cannot be written in closed form. Therefore, the modified MLE is proposed using the technique replaced some terms in the maximum likelihood equations by their expectations. The newly modified MLE using LRSS is shown to be superior to the MLE using SRS. Finally, the proposed method is applied to a real data set on metastatic renal carcinoma study.     

Estimation for Continuous Branching Processes

The maximum-likelihood estimator for the curved exponential family given by continuous branching processes with immigration is investigated. These processes originated from population biology but also model the dynamics of interest rates and development of the state of technology in economics. It is proved that in contrast to branching processes with discrete space and/or time the MLE gives a unified approach to the inference. In order to include singular subdomains of the parameter space we modify the MLE slightly. Consistency and asymptotic normality for the MLE are considered. Concerning the asymptotic theory of the experiments, all three properties LAQ, LAN, and LAMN occur for different submodels     

Asymptotic properties of the maximum-likelihood estimator in zero-inflated binomial regression

The zero-inflated binomial (ZIB) regression model was proposed to account for excess zeros in binomial regression. Since then, the model has been applied in various fields, such as ecology and epidemiology. In these applications, maximum-likelihood estimation (MLE) is used to derive parameter estimates. However, theoretical properties of the MLE in ZIB regression have not yet been rigorously established. The current paper fills this gap and thus provides a rigorous basis for applying the model. Consistency and asymptotic normality of the MLE in ZIB regression are proved. A consistent estimator of the asymptotic variance–covariance matrix of the MLE is also provided. Finite-sample behavior of the estimator is assessed via simulations. Finally, an analysis of a data set in the field of health economics illustrates the paper.     

时变系数空间自回归面板数据模型的极大似然估计

本文对扰动项存在跨时期的异方差、但不存在序列相关的时变系数空间自回归模型提出了极大似然的估计方法,并证明了该估计量的一致性,同时,证明了该估计量渐进服从正态分布,由此说明该估计量具有优良的大样本性质。同时,我们还对本文所提出估计量的小样本性质进行了数值模拟。本文研究表明,估计量虽然在N较小时偏差较大,但是随着N的不断增加,估计量偏差减小,体现了比较优良的渐进性质。同时,估计量的偏差会随着时期数的增加而变大,这说明本文所提出的估计方法适用于个体数较多、时期数较少的短面板数据。     

Mixed Liu estimator in linear measurement error models

In this paper, we introduce mixed Liu estimator (MLE) for the vector of parameters in linear measurement error models by unifying the sample and the prior information. The MLE is a generalization of the mixed estimator (ME) and Liu estimator (LE). In particular, asymptotic normality properties of the estimators are discussed, and the performance of the MLE over the LE and ME are compared based on mean squared error matrix (MSEM). Finally, a Monte Carlo simulation and a numerical example are also presented for analysis.     

On cusp location estimation for perturbed dynamical systems

We consider the problem of parameter estimation in the case of observation of the trajectory of the diffusion process. We suppose that the drift coefficient has a singularity of cusp type and that the unknown parameter corresponds to the position of the point of the cusp. The asymptotic properties of the maximum likelihood estimator and Bayesian estimators are described in the asymptotic of small noise, that is, as the diffusion coefficient tends to zero. The consistency, limit distributions, and the convergence of moments of these estimators are established.     

Inference in a generalized endpoint-inflated binomial regression model

Generalized endpoint-inflated binomial regression was recently proposed to model count data with large frequencies of both zeros and right-endpoints. Maximum likelihood estimation (MLE) was developed for this model and simulations suggest that the resulting estimates behave well. However, large-sample properties of the MLE have not yet been rigorously established. Such results are however essential for ensuring reliable statistical inference and decision-making. This paper addresses this issue. Identifiability of the generalized endpoint-inflated binomial regression model is first proved. Then, consistency and asymptotic normality of the MLE are established. A simulation study is conducted to assess finite-sample behaviour of the estimator.     

Variance estimation for tree-order restricted models

In this article, we discuss the estimation of the common variance of several normal populations with tree-order restricted means. We discuss the asymptotic properties of the maximum-likelihood estimator (MLE) of the variance as the number of populations tends to infinity. We consider several cases of various orders of the sample sizes and show that the MLE of the variance may or may not be consistent or be asymptotically normal.     

Variable Selection for Partially Linear Models with Randomly Censored Data

This article proposes a variable selection procedure for partially linear models with right-censored data via penalized least squares. We apply the SCAD penalty to select significant variables and estimate unknown parameters simultaneously. The sampling properties for the proposed procedure are investigated. The rate of convergence and the asymptotic normality of the proposed estimators are established. Furthermore, the SCAD-penalized estimators of the nonzero coefficients are shown to have the asymptotic oracle property. In addition, an iterative algorithm is proposed to find the solution of the penalized least squares. Simulation studies are conducted to examine the finite sample performance of the proposed method.     

Aalen's linear model for doubly censored data

Double censoring often occurs in registry studies when left censoring is present in addition to right censoring. In this work, we examine estimation of Aalen's nonparametric regression coefficients based on doubly censored data. We propose two estimation techniques. The first type of estimators, including ordinary least squared (OLS) estimator and weighted least squared (WLS) estimators, are obtained using martingale arguments. The second type of estimator, the maximum likelihood estimator (MLE), is obtained via expectation-maximization (EM) algorithms that treat the survival times of left censored observations as missing. Asymptotic properties, including the uniform consistency and weak convergence, are established for the MLE. Simulation results demonstrate that the MLE is more efficient than the OLS and WLS estimators.     

Inference of R=P[X<Y] for generalized logistic distribution

This paper deals with the estimation of R=P[X<Y] when X and Y come from two independent generalized logistic distributions with different parameters. The maximum-likelihood estimator (MLE) and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Assuming that the common scale parameter is known, the MLE, uniformly minimum variance unbiased estimator, Bayes estimation and confidence interval of R are obtained. The MLE of R, asymptotic distribution of R in the general case, is also discussed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.     

Minimum disparity estimation based on combined disparities: Asymptotic results

The maximum likelihood estimator (MLE) is asymptotically efficient for most parametric models under standard regularity conditions, but it has very poor robustness properties. On the other hand some of the minimum disparity estimators like the minimum Hellinger distance estimator (MHDE) have strong robustness features but their small sample efficiency at the model turns out to be very poor compared to the MLE. Methods based on the minimization of some combined disparities can substantially improve their small sample performances without affecting their robustness properties (Park et al., 1995). All studies involving the combined disparity have so far been empirical, and there are no results on the asymptotic properties of these estimators. In view of the usefulness of these procedures this is a major gap in theory, which we try to fill through the present work. Some illustrations of the performance of the estimators and the corresponding tests are also provided.     

The efficiency of a test based on the asymptotic distribution of the mle for a linear functional relationship

A rigorous derivation is given of the asymptotic normality of the MLE of a linear functional relationship. Using these results, it is shown that the test proposed by VILLEGAS (1964) has Pitman efficiency zero w.r.t, a test based on the asymptotic distribution of the MLE.     

Fisher Information for Fractional Brownian Motion Under High-Frequency Discrete Sampling

We investigate the issue of the validation of the local asymptotic normality property of three characterizing parameters of the fractional Brownian motion under high-frequency discrete sampling. We prove that the local asymptotic normality property holds true for the likelihood only when at least one of the volatility parameter and the Hurst exponent is known. We provide optimal rates of convergence of the three parameters and Fisher information matrix in closed form.