Robust Estimators for the Parameters of Multivariate Lognormal Distribution

Abstract

In this paper, we introduce a class of location and scale estimators for the p-variate lognormal distribution. These estimators are obtained by applying a log transform to the data, computing robust Fisher consistent estimators for the obtained Gaussian data and transforming those estimators for the lognormal using the relationship between the parameters of both distributions. We prove some of the properties of these estimators, such as Fisher consistency, robustness and asymptotic normality.     

On Optimal Estimating Functions in the Presence of Nuisance Parameters

When there is only one interesting parameter θ₁ and one nuisance parameter θ₂, Godambe and Thompson (1974 Godambe , V. P. , Thompson , M. E. ( 1974 ). Estimating equations in the presence of a nuisance parameter . Ann. Statist. 2 : 568 – 571 .[Crossref], [Web of Science ®] , [Google Scholar]) showed that the optimal estimating function for θ₁ essentially is a linear function of the θ₁-score, the square of the θ₂-score, and the derivative of θ₂-score with respect to θ₂. Mukhopadhyay (2000b) generalized this result to m nuisance parameters. Mukhopadhyay (2000 Mukhopadhyay , P. ( 2000 ). On some lower bounds for the variance of an estimating function . Int. J. Math. Statist. Sci. 9 ( 2 ). [Google Scholar] 2002a Mukhopadhyay , P. ( 2002a ). On estimating functions in the presence of a nuisance parameter . Commun. Statist. Theor. Mem. 31 ( 1 ): 31 – 36 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar] b Mukhopadhyay , P. ( 2002b ): Some lower bounds on variance of estimating functions . J. Statist. Res. 36 ( 2 ): 189 – 197 . [Google Scholar]) obtained lower bounds to the variance of regular estimating functions in the presence of nuisance parameters. Taking cues from these results we propose a method of finding optimal estimating function for θ₁ by taking the multiple regression equation on θ₁ score and Bhattacharyya's (1946 Bhattacharyya , A. ( 1946 ). On some analogues of the amount of information and their use in statistical estimation . Sankhya A 8 : 1 – 14 . [Google Scholar]) scores with respect to θ₂. The result is extended to the case of m nuisance parameters.     

Nuisance Parameters and the Use of Exploratory Graphical Methods in a Bayesian Analysis

Consider the problem of inference about a parameter θ in the presence of a nuisance parameter v. In a Bayesian framework, a number of posterior distributions may be of interest, including the joint posterior of (θ, ν), the marginal posterior of θ, and the posterior of θ conditional on different values of ν. The interpretation of these various posteriors is greatly simplified if a transformation (θ, h(θ, ν)) can be found so that θ and h(θ, v) are approximately independent. In this article, we consider a graphical method for finding this independence transformation, motivated by techniques from exploratory data analysis. Some simple examples of the use of this method are given and some of the implications of this approximate independence in a Bayesian analysis are discussed.     

On Maximum Likelihood Estimation for the Two Parameter Burr XII Distribution

This article considers three related aspects of maximum likelihood estimation of parameters in the two-parameter Burr XII distribution. Specifically, we first provide further clarification to some limiting results in Wingo (1993 Wingo , D. R. ( 1993 ). Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring . Microelectron. Reliab. 33 : 1251 – 1257 .[Crossref], [Web of Science ®] , [Google Scholar]). We then focus on details in a proof of the uniqueness of the maximum likelihood estimators. Finally, we consider using the likelihood approach for data which does not satisfy Wingo's criterion, and show that this results in fitting either a Pareto distribution or an intuitively sensible degenerate distribution to the data. The discussion here is completely general, and not restricted to data obtained under Type II censoring.     

Corrections for Bias in Maximum Likelihood Parameter Estimates Due to Nuisance Parameters

Abstract

In his Fisher Lecture, Efron (Efron, B. R. A. (1998 Efron, B. R. A. 1998. Fisher in the 21st century (with discussion). Statistical Science, 13: 95–122. [Crossref], [Web of Science ®] , [Google Scholar]). Fisher in the 21st Century (with discussion). Statistical Science 13:95–122) pointed out that maximum likelihood estimates (MLE) can be badly biased in certain situations involving many nuisance parameters. He predicted that with modern computing equipment a computer-modified version of the MLE that was less biased could become the default estimator of choice in applied problems in the 21st century. This article discusses three modifications—Lindsay's conditional likelihood, integrated likelihood, and Bartlett's bias-corrected estimating function. Each is evaluated through a study of the bias and MSE of the estimates in a stratified Weibull model with a moderate number of nuisance parameters. In Lindsay's estimating equation, three different methods for estimation of the nuisance parameters are evaluated—the restricted maximum likelihood estimate (RMLE), a Bayes estimator, and a linear Bayes estimator. In our model, the conditional likelihood with RMLE of the nuisance parameters is equivalent to Bartlett's bias-corrected estimating function. In the simulation we show that Lindsay's conditional likelihood is in general preferred, irrespective of the estimator of the nuisance parameters. Although the integrated likelihood has smaller MSE when the precise nature of the prior distribution of the nuisance parameters is known, this approach may perform poorly in cases where the prior distribution of the nuisance parameters is not known, especially using a non-informative prior. In practice, Lindsay's method using the RMLE of the nuisance parameters is recommended.     

Estimation of parameters of the gped

Janardan (1973) introduced the generalized Polya-Eggenberger distribution as a limiting form of the generalized Markov-Polya distribution (GMPD), Ja¬nardan (1998) derived GPED formally by means of Lagrange's expansion and discussed its various properties systematically. Here, a new urn model is pro¬vided for the GPED. Moment estimators of the parameters are given in closed form. Maximum hkelihood estimators are also given. Some apphcations are provided.     

On the Estimation of Parameters of Kumaraswamy-G Distributions

Cordeiro and de Castro proposed a new family of generalized distributions based on the Kumaraswamy distribution (denoted as Kw-G). Nadarajah et al. showed that the density function of the new family of distributions can be expressed as a linear combination of the density of exponentiated family of distributions. They derived some properties of Kw-G distributions and discussed estimation of parameters using the maximum likelihood (ML) method. Cheng and Amin and Ranneby introduced a new method of estimating parameters based on Kullback–Leibler divergence (the maximum spacing (MSP) method). In this article, the estimates of parameters of Kw-G distributions are obtained using the MSP method. For some special Kw-G distributions, the new estimators are compared with ML estimators. It is shown by simulations and a real data application that MSP estimators have better properties than ML estimators.     

Orthogonalized Residuals for Estimation of Marginally Specified Association Parameters in Multivariate Binary Data

Abstract. This paper focuses on marginal regression models for correlated binary responses when estimation of the association structure is of primary interest. A new estimating function approach based on orthogonalized residuals is proposed. A special case of the proposed procedure allows a new representation of the alternating logistic regressions method through marginal residuals. The connections between second‐order generalized estimating equations, alternating logistic regressions, pseudo‐likelihood and other methods are explored. Efficiency comparisons are presented, with emphasis on variable cluster size and on the role of higher‐order assumptions. The new method is illustrated with an analysis of data on impaired pulmonary function.     

Quantile Estimation in the Presence of Auxiliary Information under Negatively Associated Samples

In this article, we apply the empirical likelihood technique to propose a new class of quantile estimators in the presence of some auxiliary information under negatively associated samples. It is shown that the proposed quantile estimators are asymptotically normally distributed with smaller asymptotic variances than those of the usual quantile estimators. It is also shown that blocking technique is an useful tool in estimating asymptotic variance under negatively associated samples, which makes it possible to construct normal approximation based confidence intervals for quantiles.     

超越对数生产函数的半参数变系数估计模型

提出超越对数生产函数的半参数变系数模型,利用Profile方法给出产出弹性函数系数的局部加权最小二乘估计,并利用非参数条件自助法对有限样本的近似分布进行模拟,给出相对精确的广义似然比检验。规模报酬约束下中国1953--2008年的实证结果拒绝超越对数生产函数模型假设,产出弹性不可简单线性化而是对数劳均资本的非线性函数,时变资本弹性表现为倒U型变化趋势,时变劳动力弹性表现为U型变化趋势。     

Classical Estimator of Parameters of the Gamma Case With Presence of Two Outliers

The authors derive the moment, maximum likelihood, and mixture estimators of parameters of the gamma distribution with presence of two outliers generated from uniform distribution. These estimators are compared empirically when all the parameters are unknown; their bias and mean squared error are investigated with the help of numerical technique. The authors shown that these estimators are asymptotically unbiased. At the end, they conclude that mixture estimators are better than the maximum likelihood and moment estimators.     

Profile Likelihood Inferences on the Partially Linear Model with a Diverging Number of Parameters

In this article, we study the profile likelihood estimation and inference on the partially linear model with a diverging number of parameters. Polynomial splines are applied to estimate the nonparametric component and we focus on constructing profile likelihood ratio statistic to examine the testing problem for the parametric component in the partially linear model. Under some regularity conditions, the asymptotic distribution of profile likelihood ratio statistic is proposed when the number of parameters grows with the sample size. Numerical studies confirm our theory.     

Box-Cox变换下联合均值与方差模型的极大似然估计

在提出Box-Cox变换下联合均值与方差模型的基础上,研究了该模型参数的估计问题.同时利用截面极大似然估计方法对变换参数λ进行估计,并对均值模型和方差模型的参数进行极大似然估计.通过随机模拟和实例研究,结果表明该模型和方法是有效和可行的.     

Robust Estimation of Parameters in Simple Linear Profiles Using M-Estimators

In many situations, the quality of a process or product may be better characterized and summarized by a relationship between the response variable and one or more explanatory variables. Parameter estimation is the first step in constructing control charts. Outliers may hamper proper classical estimators and lead to incorrect conclusions. To remedy the problem of outliers, robust methods have been developed recently. In this article, a robust method is introduced for estimating the parameters of simple linear profiles. Two weight functions, Huber and Bisquare, are applied in the estimation algorithm. In addition, a method for robust estimation of the error terms variance is proposed. Simulation studies are done to investigate and evaluate the performance of the proposed estimator, as well as the classical one, in the presence and absence of outliers under different scenarios by the means of MSE criterion. The results reveal that the robust estimators proposed in this research perform as well as classical estimators in the absence of outliers and even considerably better when outliers exist. The maximum value of variance estimate in one scenario obtained from classical estimator is 10.9, while this value is 1.66 and 1.27 from proposed robust estimators when its actual value is 1.     

Estimation of the Parameters of a Clumped Binomial Model via the EM Algorithm

Binary-response data arise in teratology and mutagenicity studies in which each treatment is applied to a group of litters. In a large experiment, a contingency table can be constructed to test the treatment X litter size interaction (see Kastenbaum and Lamphiear 1959). In situations in which there is a clumped category, as in the Kastenbaum and Lamphiear mice-depletion data, a clumped binomial model (Koch et al. 1976) or a clumped beta-binomial model (Paul 1979) can be used to analyze these data. When a clumped binomial model is appropriate, the maximum likelihood estimates of the parameters of the model under the hypothesis of no treatment X litter size interaction, as well as under the hypothesis of the said interaction, can be estimated via the EM algorithm for computing maximum likelihood estimates from incomplete data (Dempster et al. 1977). In this article the EM algorithm is described and used to test treatment X litter size interaction for the Kastenbaum and Lamphiear data and for a set of data given in Luning et al. (1966).     

A Note on Consistent Estimation of Multivariate Parameters in Ergodic Diffusion Models

Certain aspects of maximum likelihood estimation for ergodic diffusions are studied via recently developed empirical process theory for martingales. This approach enables us to remove some undesirable regularity conditions that usually appear in the statistical literature on ergodic diffusions. In particular, dimension dependent conditions for the existence of a continuous likelihood and for consistency of the maximum likelihood estimator turn out to be unnecessary.     

Estimation and Prediction in the Presence of Spatial Confounding for Spatial Linear Models

In studies that produce data with spatial structure, it is common that covariates of interest vary spatially in addition to the error. Because of this, the error and covariate are often correlated. When this occurs, it is difficult to distinguish the covariate effect from residual spatial variation. In an i.i.d. normal error setting, it is well known that this type of correlation produces biased coefficient estimates, but predictions remain unbiased. In a spatial setting, recent studies have shown that coefficient estimates remain biased, but spatial prediction has not been addressed. The purpose of this paper is to provide a more detailed study of coefficient estimation from spatial models when covariate and error are correlated and then begin a formal study regarding spatial prediction. This is carried out by investigating properties of the generalized least squares estimator and the best linear unbiased predictor when a spatial random effect and a covariate are jointly modelled. Under this setup, we demonstrate that the mean squared prediction error is possibly reduced when covariate and error are correlated.     

Estimation of Mittag-Leffler Parameters

We propose a procedure for estimating the parameters of the Mittag-Leffler (ML) and the generalized Mittag-Leffler (GML) distributions. The algorithm is less restrictive, computationally simple, and necessary to make these models usable in practice. A comparison with the fractional moment estimator indicated favorable results for the proposed method.     

On the Empirical Likelihood Ratio for Smooth Functions of M-functionals

It is known that the empirical likelihood ratio can be used to construct confidence regions for smooth functions of the mean, Fréchet differentiable statistical functionals and for a class of M-functionals. In this paper, we argue that this use can be extended to the class of functionals which are smooth functions of M-functionals. In particular, we find the conditions under which the empirical log-likelihood ratio for this kind of functionals admits a χ² approxima tion. Furthermore, we investigate, by simulation methods, the related approximation error in some contexts of practical interest.     

Prediction Error Estimation Under Bregman Divergence for Non-Parametric Regression and Classification

Abstract.  Prediction error is critical to assess model fit and evaluate model prediction. We propose the cross-validation (CV) and approximated CV methods for estimating prediction error under the Bregman divergence (BD), which embeds nearly all of the commonly used loss functions in the regression, classification procedures and machine learning literature. The approximated CV formulas are analytically derived, which facilitate fast estimation of prediction error under BD. We then study a data-driven optimal bandwidth selector for local-likelihood estimation that minimizes the overall prediction error or equivalently the covariance penalty. It is shown that the covariance penalty and CV methods converge to the same mean-prediction-error-criterion. We also propose a lower-bound scheme for computing the local logistic regression estimates and demonstrate that the algorithm monotonically enhances the target local likelihood and converges. The idea and methods are extended to the generalized varying-coefficient models and additive models.