Multiple Hypothesis Testing for Variable Selection

We propose two new procedures based on multiple hypothesis testing for correct support estimation in high‐dimensional sparse linear models. We conclusively prove that both procedures are powerful and do not require the sample size to be large. The first procedure tackles the atypical setting of ordered variable selection through an extension of a testing procedure previously developed in the context of a linear hypothesis. The second procedure is the main contribution of this paper. It enables data analysts to perform support estimation in the general high‐dimensional framework of non‐ordered variable selection. A thorough simulation study and applications to real datasets using the R package mht shows that our non‐ordered variable procedure produces excellent results in terms of correct support estimation as well as in terms of mean square errors and false discovery rate, when compared to common methods such as the Lasso, the SCAD penalty, forward regression or the false discovery rate procedure (FDR).     

An iterative self-weighting procedure for fitting straight lines to heteroscedastic data

An iterative, self-weighting procedure is presented for the fitting of straight lines to data with heteroscedastic error-variances in the response variable. The error-variances are assumed to be unknown, even relative to each other. The procedure is compared with the “resistant line” method advocated by Emerson and Hoaglin [Emerson and Hoaglin, 1983], using extensive Monte-Carlo calculations. The proposed method is simple and easily automated, and gives parameter-estimates with smaller variance (higher efficiency) than those resulting from the resistant line technique. A BASIC program to perform the self-weighting fit is given in an appendix     

Partially linear transformation model for length-biased and right-censored data

In this paper, we consider a partially linear transformation model for data subject to length-biasedness and right-censoring which frequently arise simultaneously in biometrics and other fields. The partially linear transformation model can account for nonlinear covariate effects in addition to linear effects on survival time, and thus reconciles a major disadvantage of the popular semiparamnetric linear transformation model. We adopt local linear fitting technique and develop an unbiased global and local estimating equations approach for the estimation of unknown covariate effects. We provide an asymptotic justification for the proposed procedure, and develop an iterative computational algorithm for its practical implementation, and a bootstrap resampling procedure for estimating the standard errors of the estimator. A simulation study shows that the proposed method performs well in finite samples, and the proposed estimator is applied to analyse the Oscar data.     

A note on the satterthwaite confidence interval for a variance

When using a Satterthwaite chi-squared approximation, it is generally thought that the approximation is satisfactory when it is applied to a positive linear combination of mean squares. In this note, we describe how the Williams - Tukey idea for getting a confidence interval for the among groups variance in a random one-way model can be incorporated into Satterthwaite’s procedure for getting a confidence interval for a variance. This adjusted Satterthwaite procedure insures that his chi-squared approximation is always applied to positive linear combinations of mean squares. A small simulation is included which suggests that the adjustment to the Satterthwaite procedure is effective.     

INCOMPLETE DATA IN GENERALIZED LINEAR MODELS WITH CONTINUOUS COVARIATES

This paper proposes a method for estimating the parameters in a generalized linear model with missing covariates. The missing covariates are assumed to come from a continuous distribution, and are assumed to be missing at random. In particular, Gaussian quadrature methods are used on the E-step of the EM algorithm, leading to an approximate EM algorithm. The parameters are then estimated using the weighted EM procedure given in Ibrahim (1990). This approximate EM procedure leads to approximate maximum likelihood estimates, whose standard errors and asymptotic properties are given. The proposed procedure is illustrated on a data set.     

Statistical inference on partial linear additive models with distortion measurement errors

We consider statistical inference for partial linear additive models (PLAMs) when the linear covariates are measured with errors and distorted by unknown functions of commonly observable confounding variables. A semiparametric profile least squares estimation procedure is proposed to estimate unknown parameter under unrestricted and restricted conditions. Asymptotic properties for the estimators are established. To test a hypothesis on the parametric components, a test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we further show that its limiting distribution is a weighted sum of independent standard chi-squared distributions. A bootstrap procedure is further proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.     

A Stepwise Confidence Interval Procedure Under Unknown Variances Based on an Asymmetric Loss Function for Toxicological Evaluation

One of the most important issues in toxicity studies is the identification of the equivalence of treatments with a placebo. Because it is unacceptable to declare non‐equivalent treatments to be equivalent, it is important to adopt a reliable statistical method to properly control the family‐wise error rate (FWER). In dealing with this issue, it is important to keep in mind that overestimating toxicity equivalence is a more serious error than underestimating toxicity equivalence. Consequently asymmetric loss functions are more appropriate than symmetric loss functions. Recently Tao, Tang & Shi (2010) developed a new procedure based on an asymmetric loss function. However, their procedure is somewhat unsatisfactory because it assumes that the variances of various dose levels are known. This assumption is restrictive for some applications. In this study we propose an improved approach based on asymmetric confidence intervals without the restrictive assumption of known variances. The asymmetry guarantees reliability in the sense that the FWER is well controlled. Although our procedure is developed assuming that the variances of various dose levels are unknown but equal, simulation studies show that our procedure still performs quite well when the variances are unequal.     

A graphical procedure for distinguishing between two data analysis pitfalls

This paper presents a graphical procedure for simultaneously distinguishing between two commonly encountered data anomaliesWhen applied in the context of one anomaly, a family of parallel lines will be estimated, and when applied in the contextofthe second anomaly, a family of lines, whose members all pass through the same point, will be estimated. It is shown that the procedure can be applied effectively using samples containing as few as two hundred bivariate observations.     

Adaptively varying-coefficient spatiotemporal models

Summary.  We propose an adaptive varying-coefficient spatiotemporal model for data that are observed irregularly over space and regularly in time. The model is capable of catching possible non-linearity (both in space and in time) and non-stationarity (in space) by allowing the auto-regressive coefficients to vary with both spatial location and an unknown index variable. We suggest a two-step procedure to estimate both the coefficient functions and the index variable, which is readily implemented and can be computed even for large spatiotemporal data sets. Our theoretical results indicate that, in the presence of the so-called nugget effect, the errors in the estimation may be reduced via the spatial smoothing—the second step in the estimation procedure proposed. The simulation results reinforce this finding. As an illustration, we apply the methodology to a data set of sea level pressure in the North Sea.     

An inspection procedure for radio frequency repeaters using a multiple linear regression method

Abstract

A radio frequency (RF) repeater is a wireless electronic device that transmits signals from a base transceiver station to a mobile station. When inspecting RF repeaters, various items are required to be tested to ensure their quality. In this paper, we propose a systematic procedure for the inspection by using a multiple linear regression method. The basic idea is to predict the inspection result without real inspection. In particular, a multicollinearity problem is considered in the regression analysis. Two case studies are conducted for validating the proposed method with an RF repeater production company in Korea.     

Testing variance components in balanced linear growth curve models

It is well known that the testing of zero variance components is a non-standard problem since the null hypothesis is on the boundary of the parameter space. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold because of this null hypothesis. To circumvent this difficulty in balanced linear growth curve models, we introduce an appropriate test statistic and suggest a permutation procedure to approximate its finite-sample distribution. The proposed test alleviates the necessity of any distributional assumptions for the random effects and errors and can easily be applied for testing multiple variance components. Our simulation studies show that the proposed test has Type I error rate close to the nominal level. The power of the proposed test is also compared with the likelihood ratio test in the simulations. An application on data from an orthodontic study is presented and discussed.     

William Ramsdell Leonard 1904–1981

In this article, the corrected F statistic in the general linear model is shown to be algebraically equivalent to the corresponding statistic in the weighted least squares procedure, whenever the corrected F statistic exists. Hence, weighted least squares analysis is, in effect, a self-corrected F test procedure.     

On eliminating inferior regression models

Consider a linear regression model with [p-1] predictor variables which is taken as the "true" model.The goal Is to select a subset of all possible reduced models such that all inferior models ‘to be defined’ are excluded with a guaranteed minimum probability.A procedure is proposed for which the exact evaluation of the probability of a correct decision 1s difficult; however, 1t is shown that the probability requirement can be met for sufficiently large sample size.Monte Carlo evaluation of the constant associated with the procedure and some ways to reduce the amount of computations Involved in the Implementation of the procedure are discussed.     

Reducing errors-in-variables bias in linear regression using compact genetic algorithms

A new technique is devised to mitigate the errors-in-variables bias in linear regression. The procedure mimics a 2-stage least squares procedure where an auxiliary regression which generates a better behaved predictor variable is derived. The generated variable is then used as a substitute for the error-prone variable in the first-stage model. The performance of the algorithm is tested by simulation and regression analyses. Simulations suggest the algorithm efficiently captures the additive error term used to contaminate the artificial variables. Regressions provide further credit to the simulations as they clearly show that the compact genetic algorithm-based estimate of the true but unobserved regressor yields considerably better results. These conclusions are robust across different sample sizes and different variance structures imposed on both the measurement error and regression disturbances.     

Local Linear Kernel Regression with Long-Range Dependent Errors

This paper considers the use of a local linear kernel regression method to test whether the mean function of a sequence of long-range dependent processes has discontinuities or change-points. It proposes a non-parametric estimation procedure and then establishes an asymptotic theory for the estimation procedure. Examples, simulated and real, illustrate the estimation procedure.     

A recursive approach for estimating missing observations in an univariate time series

A linear recursive technique that does not use the Kalman filter approach is proposed to estimate missing observations in an univariate time series. It is assumed that the series follows an invertible ARIMA model. The procedure is based on the restricted forecasting approach, and the recursive linear estimators are optimal in terms of minimum mean-square error.     

Variable Selection for Semiparametric Partially Linear Covariate-Adjusted Regression Models

In this article, the partially linear covariate-adjusted regression models are considered, and the penalized least-squares procedure is proposed to simultaneously select variables and estimate the parametric components. The rate of convergence and the asymptotic normality of the resulting estimators are established under some regularization conditions. With the proper choices of the penalty functions and tuning parameters, it is shown that the proposed procedure can be as efficient as the oracle estimators. Some Monte Carlo simulation studies and a real data application are carried out to assess the finite sample performances for the proposed method.     

Transforming linear functions to normality:optimal component powers

This paper considers the analysis of linear models where the response variable is a linear function of observable component variables. For example, scores on two or more psychometric measures (the component variables) might be weighted and summed to construct a single response variable in a psychological study. A linear model is then fit to the response variable. The question addressed in this paper is how to optimally transform the component variables so that the response is approximately normally distributed. The transformed component variables, themselves, need not be jointly normal. Two cases are considered; in both cases, the Box-Cox power family of transformations is employed. In Case I, the coefficients of the linear transformation are known constants. In Case II, the linear function is the first principal component based on the matrix of correlations among the transformed component variables. For each case, an algorithm is described for finding the transformation powers that minimize a generalized Anderson-Darling statistic. The proposed transformation procedure is compared to likelihood-based methods by means of simulation. The proposed method rarely performed worse than likelihood-based methods and for many data sets performed substantially better. As an illustration, the algorithm is applied to a problem from rural sociology and social psychology; namely scaling family residences along an urban-rural dimension.     

Best Linear Classification Rule for Multivariate Interval Screened Data

In classification analysis, the target variable is often in practice defined by an underlying multivariate interval screening scheme. This engenders the problem of properly characterizing the screened populations as well as that of obtaining a classification procedure. Such problems paved the way for the development of yet another linear classification procedure and the incorporation of a class of skew-elliptical distributions for describing evolutions in the populations. To render the linear procedure effective, this article considers derivation and properties of the classification procedure as well as efficient estimation. The procedure is illustrated in applications to real and simulation data.     

Local rank estimation and related test for varying-coefficient partially linear models

This paper develops a robust estimation procedure for the varying-coefficient partially linear model via local rank technique. The new procedure provides a highly efficient and robust alternative to the local linear least-squares method. In other words, the proposed method is highly efficient across a wide class of non-normal error distributions and it only loses a small amount of efficiency for normal error. Moreover, a test for the hypothesis of constancy for the nonparametric component is proposed. The test statistic is simple and thus the test procedure can be easily implemented. We conduct Monte Carlo simulation to examine the finite sample performance of the proposed procedures and apply them to analyse the environment data set. Both the theoretical and the numerical results demonstrate that the performance of our approach is at least comparable to those existing competitors.