Robustness of dual divergence estimators for models satisfying linear constraints

We consider new classes of estimators and test statistics for models satisfying linear constraints with unknown parameter. These procedures are based on minimization of divergences through duality techniques. We prove that, for various divergences, the new approach provides robust estimation and test procedures, unlike the empirical likelihood method. We give general results using the influence function approach, which we exemplify in detail in the case of the Cressie–Read divergences. It is found that the Hellinger distance is one of the divergences that leads to robust procedures.     

广义Pareto分布的广义有偏概率加权矩估计方法

广义Pareto分布(GPD)是统计分析中一个极为重要的分布,被广泛应用于金融、保险、水文及气象等领域.传统的参数估计方法如极大似然估计、矩估计及概率加权矩估计方法等已被广泛应用,但使用中存在一定的局限性.虽然提出很多改进方法如广义概率加权矩估计、L矩和LH矩法等,但都是研究完全样本的估计问题,而在水文及气象等应用领域常出现截尾样本.本文基于概率加权矩理论,利用截尾样本对三参数GPD提出一种应用范围广且简单易行的参数估计方法,可有效减弱异常值的影响.首先求解出具有较高精度的形状参数的参数估计,其次得出位置参数及尺度参数的参数估计.通过Monte Carlo模拟说明该方法估计精度较高.     

Estimation par le minimum de distance de Hellinger dʼun processus ARFIMA

In this Note, we determine the minimum Hellinger distance estimate of an ARFIMA (AutoRegressive Fractionally Integrated Moving Average) process. The estimate minimizes the Hellinger distance between the probability density function of the innovation of the process and a parameterized random function. Under some assumptions, we establish the asymptotic properties of this estimate.     

Asymptotics for estimation and testing procedures under loss of identifiability

Statistical analyses commonly make use of models that suffer from loss of identifiability. In this paper, we address important issues related to the parameter estimation and hypothesis testing in models with loss of identifiability. That is, there are multiple parameter points corresponding to the same true model. We refer the set of these parameter points to as the set of true parameter values. We consider the case where the set of true parameter values is allowed to be very large or even infinite, some parameter values may lie on the boundary of the parameter space, and the data are not necessarily independently and identically distributed. Our results are applicable to a large class of estimators and their related testing statistics derived from optimizing an objective function such as a likelihood. We examine three specific examples: (i) a finite mixture logistic regression model; (ii) stationary ARMA processes; (iii) general quadratic approximation using Hellinger distance. The applications to these examples demonstrate the applicability of our results in a broad range of difficult statistical problems.     

A new time domain estimation of k-factors GARMA processes

We address the problem of parameter estimation of long memory time series. We consider k-factors Gegenbauer Autoregressive Moving Average (k-GARMA) processes and we estimate their parameters by the minimum Hellinger distance estimator. We establish the consistency of the estimator and the asymptotic normality for some bandwidth choice.     

On a proper way to select population failure distribution and a stochastic optimization method in parameter estimation

It is widely accepted that the Weibull distribution plays an important role in reliability applications. The reliability of a product or a system is the probability that the product or the system will still function for a specified time period when operating under some confined conditions. Parameter estimation for the three parameter Weibull distribution has been studied by many researchers in the past. Maximum likelihood has traditionally been the main method of estimation for Weibull parameters along with other recently proposed hybrids of optimization methods. In this paper, we use a stochastic optimization method called the Markov Chain Monte Carlo (MCMC) to carry out the estimation. The method is extremely flexible and inference for any quantity of interest is easily obtained.     

Minimum Hellinger distance estimation for a two-sample semiparametric cure rate model with censored survival data

Efficiency and robustness are two essential concerns on statistical estimation. Unfortunately, it was widely accepted that there existed a contradiction between achieving efficiency and robustness simultaneously. For parametric models with complete data, the minimum Hellinger distance estimation introduced by Beran (Ann Stat 5:445–463, 1977) has been shown that it can reconcile this contradiction. Because data in biostatistics, actuarial science or economics are often subject to censoring and even involve a fraction of long-term survivors, our study aims to extend the minimum Hellinger distance estimation to a two-sample semiparametric cure rate model with right-censored survival data. The asymptotic properties such as consistency, efficiency, normality, and robustness of the proposed estimator have been considered and its performances are examined via simulation studies in comparison with those of the maximum semiparametric conditional likelihood estimator introduced by Shen et al. (J Am Stat Assoc 102:1235–1244, 2007). Finally, our method is illustrated by analyzing a real data set: Bone Marrow Transplant Data.     

Minimum Hellinger distance estimation for supercritical Galton–Watson processes

This paper studies the asymptotic behavior of the minimum Hellinger distance estimator of the underlying parameter in a supercritical branching process whose offspring distribution is known to belong to a parametric family. This estimator is shown to be asymptotically normal, efficient at the true model and robust against gross errors. These extend the results of Beran (Ann. Statist. 5, 445–463 (1977)) from an i.i.d., continuous setup to a dependent, discrete setup.     

Robust Simulation-Based Estimation of ARMA Models

This article proposes a new approach to the robust estimation of a mixed autoregressive and moving average (ARMA) model. It is based on the indirect inference method that originally was proposed for models with an intractable likelihood function. The estimation algorithm proposed is based on an auxiliary autoregressive representation whose parameters are first estimated on the observed time series and then on data simulated from the ARMA model. To simulate data the parameters of the ARMA model have to be set. By varying these we can minimize a distance between the simulation-based and the observation-based auxiliary estimate. The argument of the minimum yields then an estimator for the parameterization of the ARMA model. This simulation-based estimation procedure inherits the properties of the auxiliary model estimator. For instance, robustness is achieved with GM estimators. An essential feature of the introduced estimator, compared to existing robust estimators for ARMA models, is its theoretical tractability that allows us to show consistency and asymptotic normality. Moreover, it is possible to characterize the influence function and the breakdown point of the estimator. In a small sample Monte Carlo study it is found that the new estimator performs fairly well when compared with existing procedures. Furthermore, with two real examples, we also compare the proposed inferential method with two different approaches based on outliers detection.     

Optimal robust insurance with a finite uncertainty set

Decision-makers who usually face model/parameter risk may prefer to act prudently by identifying optimal contracts that are robust to such sources of uncertainty. In this paper, we tackle this issue under a finite uncertainty set that contains a number of probability models that are candidates for the “true”, but unknown model. Various robust optimisation models are proposed, some of which are already known in the literature, and we show that all of them can be efficiently solved via Second Order Conic Programming (SOCP). Numerical experiments are run for various risk preference choices and it is found that for relatively large sample size, the modeler should focus on finding the best possible fit for the unknown probability model in order to achieve the most robust decision. If only small samples are available, then the modeler should consider two robust optimisation models, namely the Weighted Average Model or Weighted Worst-case Model, rather than focusing on statistical tools aiming to estimate the probability model. Amongst those two, the better choice of the robust optimisation model depends on how much interest the modeler puts on the tail risk when defining its objective function. These findings suggest that one should be very careful when robust optimal decisions are sought in the sense that the modeler should first understand the features of its objective function and the size of the available data, and then to decide whether robust optimisation or statistical inferences is the best practical approach.     

Robust Estimation of the Generalized Pareto Distribution

One approach used for analyzing extremes is to fit the excesses over a high threshold by a generalized Pareto distribution. For the estimation of the shape and scale parameters in the generalized Pareto distribution, under some restrictions on the value of the scale parameter, maximum likelihood, method of moments and probability weighted moments' estimators are available. However, these are not robust estimators. In this paper we implement a robust estimation procedure known as the method of medians (He and Fung, 1999) to estimate the parameters in the generalized Pareto distribution. The asymptotic distribution of our estimator is normal for any value of the shape parameter except –1.     

Copula parameter estimation by maximum-likelihood and minimum-distance estimators: a simulation study

The purpose of this paper is to present a comprehensive Monte Carlo simulation study on the performance of minimum-distance (MD) and maximum-likelihood (ML) estimators for bivariate parametric copulas. In particular, I consider Cramér-von-Mises-, Kolmogorov-Smirnov- and L ₁-variants of the CvM-statistic based on the empirical copula process, Kendall’s dependence function and Rosenblatt’s probability integral transform. The results presented in this paper show that regardless of the parametric form of the copula, the sample size or the location of the parameter, maximum-likelihood yields smaller estimation biases at less computational effort than any of the MD-estimators. The MD-estimators based on copula goodness-of-fit metrics, on the other hand, suffer from large biases especially when used for estimating the parameters of archimedean copulas. Moreover, the results show that the bias and efficiency of the minimum-distance estimators are strongly influenced by the location of the parameter. Conversely, the results for the maximum-likelihood estimator are relatively stable over the parameter interval of the respective parametric copula.     

A parametric approach to correspondence analysis

We compare correspondence analysis (CA) and the alternative approach using Hellinger distance (HD), for representing categorical data in a contingency table. As both methods may be appropriate, we introduce a parameter and define a generalized version of correspondence analysis (GCA) which contains CA and HD as particular cases. Comparison with alternative approaches are performed. We propose a coefficient which globally measures the similarity between CA and GCA, which can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions on the difference between both representations. Two criteria for choosing the best value of the parameter are proposed.     

SEM modeling with singular moment matrices Part III: GLS estimation

We discuss generalized least squares (GLS) and maximum likelihood (ML) estimation for structural equations models (SEM), when the sample moment matrices are possibly singular. This occurs in several instances, for example, for panel data when there are more panel waves than independent replications or for time series data where the number of time points is large, but only one unit is observed. In previous articles, it was shown that ML estimation of the SEM is possible by using a correct Gaussian likelihood function. In this article, the usual GLS fit function is modified so that it is also defined for singular sample moment matrices S. In large samples, GLS and ML estimation perform similarly, and the modified GLS approach is a good alternative when S becomes nearly singular. Both GLS approaches do not work for N = 1, since here S = 0 and the modified GLS approach yields biased estimates. In conclusion, ML estimation (and pseudo ML under misspecification) is recommended for all sample sizes including N = 1.     

On the Parameters Estimators for a Discrete Analog of the Generalized Exponential Distribution

Nekoukhou et. al (Commun. Statist. Th. Meth., 2012) introduced a two-parameters discrete probability distribution so-called Discrete Analog of the Generalized Exponential Distribution (in short, DGED). We shall attempt to derive conditions under which a solution for the system of likelihood equations exists and coincides with the maximum likelihood (ML) estimators of the DGED. This kind of ML estimators are coincided with some moment estimators. An approximate computation based on Fisher’s accumulation method is presented in order for the ML estimations of the unknown parameters. Simulation study is also illustrated. Meanwhile, in the sequel two special cases of the DGED are considered. Some statistical properties for such special cases of the DGED are provided. We also propose a linear regression-type model for estimation of the parameter. Finally, we fit the DGED to a real data set and compare it with two other discrete distributions.     

多个正态总体均值相等的VDR检验

本文将随机估计由一维参数扩展至多维参数,基于随机估计的密度函数提出VDR检验.在总体方差已知和未知的两种情形下,本文讨论多个正态总体均值是否相同的VDR检验过程,而且得到精确的检验.单因素方差分析是VDR检验的特例.模拟研究表明,VDR检验是一个普遍适用的方法.     

Optimal robust estimates using the Hellinger distance

Optimal robust M-estimates of a multidimensional parameter are described using Hampel’s infinitesimal approach. The optimal estimates are derived by minimizing a measure of efficiency under the model, subject to a bounded measure of infinitesimal robustness. To this purpose we define measures of efficiency and infinitesimal sensitivity based on the Hellinger distance. We show that these two measures coincide with similar ones defined by Yohai using the Kullback–Leibler divergence, and therefore the corresponding optimal estimates coincide too. We also give an example where we fit a negative binomial distribution to a real dataset of “days of stay in hospital” using the optimal robust estimates.     

Robust regression function estimation

A robust estimator of the regression function is proposed combining kernel methods as introduced for density estimation and robust location estimation techniques. Weak and strong consistency and asymptotic normality are shown under mild conditions on the kernel sequence. The asymptotic variance is a product from a factor depending only on the kernel and a factor similar to the asymptotic variance in robust estimation of location. The estimation is minimax robust in the sense of Huber (1964). Robust estimation of a location parameter. Ann. Math. Statist.33 73–101.     

On the characterization of fisher information and stability of the least favorable lattice distributions

The paper is concerned with the stability properties of the least favorable distributions minimizing the Fisher information in a given class of distributions. The derivation of a least favorable distribution (the solution of a variational problem) is a necessary stage of the Huber minimax approach in robust estimation of a location parameter. Generally, the solutions of variational problems essentially depend on the regularity restrictions of a functional class. The stability of these optimal solutions to violations of the smoothness restrictions is studied under the lattice distribution classes. The discrete analogues of Fisher information are obtained in these cases. They have the form of the Hellinger metrics with the estimation of a real continuous location parameter and the form of the X² metrics with the estimation of an integer discrete location parameter. The analytical expressions for the corresponding least favorable discrete distributions are derived in some classes of lattice distributions by means of generating functions and Bellman's recursive functional equations of dynamic programming. These classes include the class of nondegenerate distributions with a restriction on the value of the density in the center of symmetry, the class of finite distributions, and the class of contaminated distributions. The obtained least favorable lattice distributions preserve the structure of their prototypes in the continuous case. These results show the stability of robust minimax solutions under different types of transitions from the continuous distribution to the discrete one. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajduszoboszló, Hungary, 1997, Part II.     

Local Walsh-average-based estimation and variable selection for single-index models

We propose a robust estimation procedure based on local Walsh-average regression(LWR) for single-index models. Our novel method provides a root-n consistent estimate of the single-index parameter under some mild regularity conditions; the estimate of the unknown link function converges at the usual rate for the nonparametric estimation of a univariate covariate. We theoretically demonstrate that the new estimators show significant efficiency gain across a wide spectrum of non-normal error distributions and have almost no loss of efficiency for the normal error. Even in the worst case, the asymptotic relative efficiency(ARE) has a lower bound compared with the least squares(LS) estimates; the lower bounds of the AREs are 0.864 and 0.8896 for the single-index parameter and nonparametric function, respectively. Moreover, the ARE of the proposed LWR-based approach versus the ARE of the LS-based method has an expression that is closely related to the ARE of the signed-rank Wilcoxon test as compared with the t-test. In addition, to obtain a sparse estimate of the single-index parameter, we develop a variable selection procedure by combining the estimation method with smoothly clipped absolute deviation penalty; this procedure is shown to possess the oracle property. We also propose a Bayes information criterion(BIC)-type criterion for selecting the tuning parameter and further prove its ability to consistently identify the true model. We conduct some Monte Carlo simulations and a real data analysis to illustrate the finite sample performance of the proposed methods.