Bounded-leverage weights for robust regression estimators

Both the least squares estimator and M-estimators of regression coefficients are susceptible to distortion when high leverage points occur among the predictor variables in a multiple linear regression model. In this article a weighting scheme which enables one to bound the leverage values of a weighted matrix of predictor variables is proposed. Bounded-leverage weighting of the predictor variables followed by M-estimation of the regression coefficients is shown to be effective in protecting against distortion due to extreme predictor-variable values, extreme response values, or outlier-induced multieollinearites. Bounded-leverage estimators can also protect against distortion by small groups of high leverage points.     

Consistent and robust variable selection in regression based on Wald test

Selection of relevant predictor variables for building a model is an important problem in the multiple linear regression. Variable selection method based on ordinary least squares estimator fails to select the set of relevant variables for building a model in the presence of outliers and leverage points. In this article, we propose a new robust variable selection criterion for selection of relevant variables in the model and establish its consistency property. Performance of the proposed method is evaluated through simulation study and real data.     

A note on functional linear regression

This work focuses on the linear regression model with functional covariate and scalar response. We compare the performance of two (parametric) linear regression estimators and a nonparametric (kernel) estimator via a Monte Carlo simulation study and the analysis of two real data sets. The first linear estimator expands the predictor and the regression weight function in terms of the trigonometric basis, while the second one uses functional principal components. The choice of the regularization degree in the linear estimators is addressed.     

Errors-in-variables regression using estimated latent variables

This note considers a method for estimating regression parameters from the data containing measurement errors using some natural estimates of the unobserved explanatory variables. It is shown that the resulting estimator is consistent not only in the usual linear regression model but also in the probit model and regression models with censoship or truncation. However, it fails to be consistent in nonlinear regression models except for special cases.     

Errors-in-variables regression using estimated latent variables

This note considers a method for estimating regression parameters from the data containing measurement errors using some natural estimates of the unobserved explanatory variables. It is shown that the resulting estimator is consistent not only in the usual linear regression model but also in the probit model and regression models with censoship or truncation. However, it fails to be consistent in nonlinear regression models except for special cases.     

Polynomial regression with errors in the variables

A polynomial functional relationship with errors in both variables can be consistently estimated by constructing an ordinary least squares estimator for the regression coefficients, assuming hypothetically the latent true regressor variable to be known, and then adjusting for the errors. If normality of the error variables can be assumed, the estimator can be simplified considerably. Only the variance of the errors in the regressor variable and its covariance with the errors of the response variable need to be known. If the variance of the errors in the dependent variable is also known, another estimator can be constructed.     

PMSE dominance of the positive-part shrinkage estimator in a regression model with proxy variables

Consider a linear regression model with some relevant regressors are unobservable. In such a situation, we estimate the model by using the proxy variables as regressors or by simply omitting the relevant regressors. In this paper, we derive the explicit formula of predictive mean squared error (PMSE) of a general family of shrinkage estimators of regression coefficients. It is shown analytically that the positive-part shrinkage estimator dominates the ordinary shrinkage estimator even when proxy variables are used in place of the unobserved variables. Also, as an example, our result is applied to the double k-class estimator proposed by Ullah and Ullah (Double k-class estimators of coefficients in linear regression. Econometrica. 1978;46:705–722). Our numerical results show that the positive-part double k-class estimator with proxy variables has preferable PMSE performance.     

Three estimators for the poisson regression model with measurement errors

We consider two consistent estimators for the parameters of the linear predictor in the Poisson regression model, where the covariate is measured with errors. The measurement errors are assumed to be normally distributed with known error variance σ _u ² . The SQS estimator, based on a conditional mean-variance model, takes the distribution of the latent covariate into account, and this is here assumed to be a normal distribution. The CS estimator, based on a corrected score function, does not use the distribution of the latent covariate. Nevertheless, for small σ _u ² , both estimators have identical asymptotic covariance matrices up to the order of σ _u ² . We also compare the consistent estimators to the naive estimator, which is based on replacing the latent covariate with its (erroneously) measured counterpart. The naive estimator is biased, but has a smaller covariance matrix than the consistent estimators (at least up to the order of σ _u ² ).     

The mixed model for survey regression estimation

Estimation of the population mean under the regression model with random components is considered. Conditions under which the random components regression estimator is design consistent are given. It is shown that consistency holds when incorrect values are used for the variance components. The regression estimator constructed with model parameters that differ considerably from the true parameters performed well in a Monte Carlo study. Variance estimators for the regression predictor are suggested. A variance estimator appropriate for estimators constructed with a biased estimator for the between-group variance component performed well in the Monte Carlo study.     

Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.     

Small area estimation under random regression coefficient models

Statistical agencies are interested to report precise estimates of linear parameters from small areas. This goal can be achieved by using model-based inference. In this sense, random regression coefficient models provide a flexible way of modelling the relationship between the target and the auxiliary variables. Because of this, empirical best linear unbiased predictor (EBLUP) estimates based on these models are introduced. A closed-formula procedure to estimate the mean-squared error of the EBLUP estimators is also given and empirically studied. Results of several simulation studies are reported as well as an application to the estimation of household normalized net annual incomes in the Spanish Living Conditions Survey.     

Iterative regularisation in nonparametric instrumental regression

This paper considers the nonparametric regression model with an additive error that is correlated with the explanatory variables. Motivated by empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. However, the estimation of a nonparametric regression function by instrumental variables is an ill-posed linear inverse problem with an unknown but estimable operator. We provide a new estimator of the regression function that is based on projection onto finite dimensional spaces and that includes an iterative regularisation method (the Landweber–Fridman method). The optimal number of iterations and the convergence of the mean square error of the resulting estimator are derived under both strong and weak source conditions. A Monte Carlo exercise shows the impact of some parameters on the estimator and concludes on the reasonable finite sample performance of the new estimator.     

Testing parametric models in linear‐directional regression

This paper presents a goodness‐of‐fit test for parametric regression models with scalar response and directional predictor, that is, a vector on a sphere of arbitrary dimension. The testing procedure is based on the weighted squared distance between a smooth and a parametric regression estimator, where the smooth regression estimator is obtained by a projected local approach. Asymptotic behaviour of the test statistic under the null hypothesis and local alternatives is provided, jointly with a consistent bootstrap algorithm for application in practice. A simulation study illustrates the performance of the test in finite samples. The procedure is applied to test a linear model in text mining.     

On multivariate quantile regression analysis

This paper investigates the estimation of parameters in a multivariate quantile regression model when the investigator wants to evaluate the associated distribution function. It proposes a new directional quantile estimator with the following properties: (1) it applies to an arbitrary number of random variables; (2) it is equivalent to estimating the distribution function allowing for non-convex distribution contours; (3) it satisfies nice equivariance properties; (4) it has desirable statistical properties (i.e., consistency and asymptotic normality); and (5) its implementation involves a modest computational burden: our proposed estimator can be obtained by solving parametric linear programming problems. As such, this paper expands the range of applications of quantile estimation for multivariate regression models.     

Poisson regression models with errors-in-variables: implication and treatment

Overdispersion has been a common phenomenon in count data and usually treated with the negative binomial model. This paper shows that measurement errors in covariates in general also lead to overdispersion on the observed data if the true data generating process is indeed the Poisson regression. This kind of overdispersion cannot be treated using the negative binomial model, as otherwise, biases will occur. To provide consistent estimates, we propose a new type of corrected score estimator assuming that the distribution of the latent variables is known. The consistency and asymptotic normality of the proposed estimator are established. Simulation results show that this estimator has good finite sample performance. We also illustrate that the Akaike information criterion and Bayesian information criterion work well for selecting the correct model if the true model is the errors-in-variables Poisson regression.     

Linear functional regression: The case of fixed design and functional response

The authors consider the problem of simple linear regression when the exogenous and endogenous variables are functional and the design is fixed. They propose an estimator for the underlying linear operator and prove its consistency under some conditions which ensure that the design is sufficiently informative. They consider the classical calibration (or inverse regression) problem and analyze a consistent estimator. They also give a simulation study. The proposed method is not hard to implement in practice.     

Estimating the density of a possibly missing response variable in nonlinear regression

This paper considers linear and nonlinear regression with a response variable that is allowed to be “missing at random”. The only structural assumptions on the distribution of the variables are that the errors have mean zero and are independent of the covariates. The independence assumption is important. It enables us to construct an estimator for the response density that uses all the observed data, in contrast to the usual local smoothing techniques, and which therefore permits a faster rate of convergence. The idea is to write the response density as a convolution integral which can be estimated by an empirical version, with a weighted residual-based kernel estimator plugged in for the error density. For an appropriate class of regression functions, and a suitably chosen bandwidth, this estimator is consistent and converges with the optimal parametric rate n^1/2. Moreover, the estimator is proved to be efficient (in the sense of Hájek and Le Cam) if an efficient estimator is used for the regression parameter.     

A generalized inverse regression estimator in multi-univariate linear calibration

A multivariate linear calibration problem, in which response variable is multivariate and explanatory variable is univariate, is considered. In this paper a class of generalized inverse regression estimators is proposed in multi-univariate linear calibration. It includes the classical estimator and the inverse regression one (or Krutchkoff estimator). For the proposed estimator we derive the expressions of bias and mean square error (MSE). Furthermore the behavior of these characteristics is investigated through an analytical method. In addition through a numerical study we confirm the existence of a generalized inverse regression estimator to improve both the classical and the inverse regression estimators on the MSE criterion.     

Improving survey-weighted least squares regression

The weighted least squares (WLS) estimator is often employed in linear regression using complex survey data to deal with the bias in ordinary least squares (OLS) arising from informative sampling. In this paper a 'quasi-Aitken WLS' (QWLS) estimator is proposed. QWLS modifies WLS in the same way that Cragg's quasi-Aitken estimator modifies OLS. It weights by the usual inverse sample inclusion probability weights multiplied by a parameterized function of covariates, where the parameters are chosen to minimize a variance criterion. The resulting estimator is consistent for the superpopulation regression coefficient under fairly mild conditions and has a smaller asymptotic variance than WLS.     

Minimum mean squared error estimation of each individual coefficient in a linear regression model

In this paper, we derive the exact mean squared error (MSE) of the minimum MSE estimator for each individual coefficient in a linear regression model, and show a sufficient condition for the minimum MSE estimator for each individual coefficient to dominate the OLS estimator. Numerical results show that when the number of independent variables is 2 and 3, the minimum MSE estimator for each individual coefficient can be a good alternative to the OLS and Stein-rule estimators.