On statistical transform methods and their efficiency

General conditions for the asymptotic efficiency of certain new inference procedures based on empirical transform functions are developed. A number of important processes, such as the empirical characteristic function, the empirical moment generating function, and the empirical moments, are considered as special cases.     

ROBUST TESTS BASED ON THE SAMPLE CHARACTERISTIC FUNCTION

A new method is described for robust analysis of variance in the balanced fixed effects case. The method uses the empirical characteristic function of the treatment samples, and has an interpretation in terms of S-estimators. The test statistic, under the null hypothesis, asymptotically follows a central chi-square distribution, and under contiguous alternatives a noncentral chi-square distribution. A Monte Carlo study suggests that, for finite samples, this is reasonably well approximated by the usual F distribution used in analysis of variance. The test statistic has a bounded influence function. The new procedure competes well with Huber's and a Wald-type procedure except in very heavy-tailed cases.     

Reducing the variance by smoothing

In this paper we show that versions of statistical functionals which are obtained by smoothing the corresponding empirical d.f. with an appropriate kernel can reduce the variance and the mean square error of the statistic. This is shown by studying the influence function of the functional. The smaller variance is achieved when the influence function is either discontinuous or piecewise linear with convexity towards the x-axis. Examples for M- and L-estimators are given.     

Nonparametrio bayes and empirical bayes estimation of the residual survival function at age t

Nonparametric Bayes and empirical Bayes estimations of the

survival function of a unit of age t (> 0) using Dirichlet

process prior are presented. The proposed empirical Bayes

estimators are found to be “asymptotically optimal” in the sense of Robbins (1955). The performances of the proposed

empirical Bayes estimators are compared with those of certain

rival estimators in terms of relative savings loss, The exact

expressions for Bayes risks are also provided in certain cases.     

INFLUENCE FUNCTION OF THE LIKELIHOOD RATIO TEST STATISTIC FOR MULTIVARIATE NORMAL SAMPLE

ABSTRACT

We derive the influence function of the likelihood ratio test statistic for multivariate normal sample. The derived influence function does not depend on the influence functions of the parameters under the null hypothesis. So we can obtain directly the empirical influence function with only the maximum likelihood estimators under the null hypothesis. Since the derived formula is a general form, it can be applied to influence analysis on many statistical testing problems.     

Influence diagnostics for censored regression models with autoregressive errors

Observations collected over time are often autocorrelated rather than independent, and sometimes include observations below or above detection limits (i.e. censored values reported as less or more than a level of detection) and/or missing data. Practitioners commonly disregard censored data cases or replace these observations with some function of the limit of detection, which often results in biased estimates. Moreover, parameter estimation can be greatly affected by the presence of influential observations in the data. In this paper we derive local influence diagnostic measures for censored regression models with autoregressive errors of order p (hereafter, AR(p)‐CR models) on the basis of the Q‐function under three useful perturbation schemes. In order to account for censoring in a likelihood‐based estimation procedure for AR(p)‐CR models, we used a stochastic approximation version of the expectation‐maximisation algorithm. The accuracy of the local influence diagnostic measure in detecting influential observations is explored through the analysis of empirical studies. The proposed methods are illustrated using data, from a study of total phosphorus concentration, that contain left‐censored observations. These methods are implemented in the R package ARCensReg.     

The influence of observations on misclassification probability in multiple discriminant analysis

The influence of observations in estimating the misclassification probability in multiple discriminant analysis is studied using the common omission approach. An empirical influence function for the misclassification probability is also derived, It can give a very good approximation to the omission approach, but the computational load is much reduced, Various extensions of the measures are suggested. The proposed measures are applied to the famous Iris data set. The same three observations are identified as having the most influence under different measures.     

Extended linear empirical bayes estimation

In this paper, the linear empirical Bayes estimation method, which is based on approximation of the Bayes estimator by a linear function, is generalized to an extended linear empirical Bayes estimation technique which represents the Bayes estimator by a series of algebraic polynomials. The extended linear empirical Bayes estimators are elaborated in the case of a location or a scale parameter. The theory is illustrated by examples of its application to the normal distribution with a location parameter and the gamma distribution with a scale parameter. The linear and the extended linear empirical Bayes estimators are constructed in these two cases and, then, studied numerically via Monte Carlo simulations. The simulations show that the extended linear empirical Bayes estimators have better convergence rates than the traditional linear empirical Bayes estimators.     

Empirical likelihood inference for general transformation models with right censored data

In this work, we consider empirical likelihood inference for general transformation models with right censored data. The models are a class of flexible semiparametric survival models and include many popular survival models as their special cases. Based on the marginal likelihood function, we define an empirical likelihood ratio statistic. Under some regularity conditions, we show that the empirical likelihood ratio statistic asymptotically follows a standard chi-squared distribution. Through some simulation studies and a real data application, we show that our proposed procedure can work fairly well even for relatively small sample size and high censoring.     

带线性约束的多元线性回归模型参数估计

本文首先构造线性约束条件下的多元线性回归模型的样本似然函数,利用Lagrange法证明其合理性。其次,从似然函数的角度讨论线性约束条件对模型参数的影响,对由传统理论得出的参数估计作出贝叶斯与经验贝叶斯的改进。做贝叶斯改进时,将矩阵正态-Wishart分布作为模型参数和精度阵的联合共轭先验分布,结合构造的似然函数得出参数的后验分布,计算出参数的贝叶斯估计;做经验贝叶斯改进时,将样本分组,从方差的角度讨论由子样得出的参数估计对总样本的参数估计的影响,计算出经验贝叶斯估计。最后,利用Matlab软件生成的随机矩阵做模拟。结果表明,这两种改进后的参数估计均较由传统理论得出的参数估计更精确,拟合结果的误差比更小,可信度更高,在大数据的情况下,这种计算方法的速度更快。     

Random Weighting Empirical Distribution Function and its Applications to Goodness-of-Fit Testing

In this article, a randomized estimator of the empirical distribution function (EDF) called random weighting empirical distribution function (RWEDF) is introduced, one special case of which is just equivalent to the Bayesian bootstrap. The consistency of the RWEDF is established under certain conditions. By substituting this new EDF for the classical EDF, we obtain new versions of some EDF test statistics for goodness-of-fit. The simulation results show that the new tests are more powerful than the corresponding tests based on the classical EDF under some cases.     

Robust m-estimators

This paper provides a summary of the influence function approach to robust estimation of parametric models. Hampel's optimality results for M-estimators with a bounded influence function is generalized to allow for arbitrary choices of the asymptotic efficiency criterion and the norm of the influence function. Further extensions to other cases of practical interest are also considered.     

Local influence assessment in canonical correlation analysis

The local influence method is adapted to canonical correlation analysis for the purpose of investigating the influence of observations. We consider a perturbation based on the empirical distribution function. An illustrative example is given to show the effectiveness of the local influence method for the identification of influential observations.     

Hazard-based nonparametric survivor function estimation

Summary.  A representation is developed that expresses the bivariate survivor function as a function of the hazard function for truncated failure time variables. This leads to a class of nonparametric survivor function estimators that avoid negative mass. The transformation from hazard function to survivor function is weakly continuous and compact differentiable, so that such properties as strong consistency, weak convergence to a Gaussian process and bootstrap applicability for a hazard function estimator are inherited by the corresponding survivor function estimator. The set of point mass assignments for a survivor function estimator is readily obtained by using a simple matrix calculation on the set of hazard rate estimators. Special cases arise from a simple empirical hazard rate estimator, and from an empirical hazard rate estimator following the redistribution of singly censored observations within strips. The latter is shown to equal van der Laan's repaired nonparametric maximum likelihood estimator, for which a Greenwood-like variance estimator is given. Simulation studies are presented to compare the moderate sample performance of various nonparametric survivor function estimators.     

Sensitivity Coefficient in Principal Component Analysis: Robust Case

In the classical principal component analysis (PCA), the empirical influence function for the sensitivity coefficient ρ is used to detect influential observations on the subspace spanned by the dominants principal components. In this article, we derive the influence function of ρ in the case where the reweighted minimum covariance determinant (MCD¹) is used as estimator of multivariate location and scatter. Our aim is to confirm the reliability in terms of robustness of the MCD¹ via the approach based on the influence function of the sensitivity coefficient.     

Empirical likelihood-based inference for parameter and nonparametric function in partially nonlinear models

This paper is concerned with statistical inference for partially nonlinear models. Empirical likelihood method for parameter in nonlinear function and nonparametric function is investigated. The empirical log-likelihood ratios are shown to be asymptotically chi-square and then the corresponding confidence intervals are constructed. By the empirical likelihood ratio functions, we also obtain the maximum empirical likelihood estimators of the parameter in nonlinear function and nonparametric function, and prove the asymptotic normality. A simulation study indicates that, compared with normal approximation-based method and the bootstrap method, the empirical likelihood method performs better in terms of coverage probabilities and average length/widths of confidence intervals/bands. An application to a real dataset is illustrated.     

Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

INFLUENCE DIAGNOSTICS FOR THE NORMAL LINEAR MODEL WITH CENSORED DATA

Methods of detecting influential observations for the normal model for censored data are proposed. These methods include one-step deletion methods, deletion of observations and the empirical influence function. Emphasis is placed on assessing the impact that a single observation has on the estimation of coefficients of the model. Functions of the coefficients such as the median lifetime are also considered. Results are compared when applied to two sets of data.     

Distance equalizers

Distance equalizers are introduced as empirical measures of central tendency that make distances to univariate data as similar as possible. These measures are made precise by means of various so-called fluctuation functions which account for distances in different ways. Distance equalizers differ from the mean as well as from the median. Also, distance equalizers relate to dispersion measures. Algorithms and closed-form solutions for special cases are given. Some computations require to perform multiextremal function minimization. Distance equalization is extendable to data from higher dimensions and to function quantization in signal processing.     

Empirical likelihood inferences for varying coefficient partially nonlinear models

In this article, empirical likelihood inferences for the varying coefficient partially nonlinear models are investigated. An empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function part and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component are proposed. The corresponding Wilks phenomena are proved and the confidence regions for parametric component and nonparametric component are constructed. Simulation studies indicate that, in terms of coverage probabilities and average areas of the confidence regions, the empirical likelihood method performs better than the normal approximation-based method. Furthermore, a real data set application is also provided to illustrate the proposed empirical likelihood estimation technique.