Conditional absolute mean calibration for partial linear multiplicative distortion measurement errors models

This paper considers partial linear single-index regression models when all the variables are measured with multiplicative distortion measurement errors. To eliminate the effect caused by the distortion, we propose the conditional absolute mean calibration. This method avoids to use the nonzero expectation conditions imposed on the variables in the literature. Using the calibrated variables, a profile least squares estimator is obtained. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. A smoothly clipped absolute deviation penalty is employed to select the relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.

This paper studies the measure of symmetry or asymmetry of a continuous variable under the multiplicative distortion measurement errors setting. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. First, two direct plug-in estimation procedures are proposed, and the empirical likelihood based confidence intervals are constructed to measure the symmetry or asymmetry of the unobserved variable. Next, we propose four test statistics for testing whether the unobserved variable is symmetric or not. The asymptotic properties of the proposed estimators and test statistics are examined. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.

In this study, we consider the testing problem about the null hypothesis that the nonlinear part in the partial linear regression model with missing response at random is a parametric function or not against the alternative that it is nonparametric. By imputation and inverse probability weighting methods, we then construct two completed data sets. Two empirical process-based tests, from these completed data sets, are introduced. Under the null hypothesis and local alterative hypotheses, the limiting null distributions and power study of the test statistics are, respectively, investigated. A nonparametric Monte Carlo test procedure, which can automatically make the test procedure scale-invariant even when the test statistics are not scale-invariant, is applied to approximate the limiting null distributions of the test statistics. Simulation study is carried out to examine the performance of the tests. We illustrate the proposed method with a real data set on monozygotic twins.

A semiparametric hypothesis testing procedure for the ROC curve area under a density ratio model

In this paper, estimation of the linear regression\nmodel is made by ordinary least squares method and the\npartially linear regression model is estimated by penalized\nleast squares method using smoothing spline. Then, it is\ninvestigated that differences and similarity in the sum of\nsquares related for linear regression and partial linear\nregression models (semi-parametric regression models). It is\ndenoted that the sum of squares in linear regression is reduced\nto sum of squares in partial linear regression models.\nFurthermore, we indicated that various sums of squares in the\nlinear regression are similar to different deviance statements in\npartial linear regression. In addition to, coefficient of the\ndetermination derived in linear regression model is easily\ngeneralized to coefficient of the determination of the partial\nlinear regression model. For this aim, it is made two different\napplications. A simulated and a real data set are considered to\nprove the claim mentioned here. In this way, this study is\nsupported with a simulation and a real data example.

Hypothesis tests

This article considers the analysis of clustered data via partial linear regression models. Adopting the idea of modeling the within-cluster correlation from the method of generalized estimating equations, a least squares type estimate of the slope parameter is obtained through piecewise local polynomial approximation of the nonparametric component. This slope estimate has several advantages: (a) It attains n1/2-consistency without undersmoothing; (b) it is efficient when correct within-cluster correlation is used, assuming multivariate normality of the error; (c) the preceding properties hold regardless of whether or not the nonparametric component is of cluster level; and (d) this estimation method naturally extends to deal with generalized partial linear models. Simulation studies and a real example are presented in support of the theory.

Over the years several discrepancy functions have been introduced both in the literature and in the software of Structural Equation Modeling (SEM). The test statistics for the discrepancy functions associated with Maximum Likelihood (ML), Generalized Least Squares (GLS), and Normal Theory Weighted Least Squares (NWLS) are all asymptotically equivalent. These test statistics are all approximately distributed as central chi-square under correct model specification and if the observed variables are multivariate normally distributed. However, it is known that the distribution of these test statistics will not approximate a central Chi-square distribution for models containing specification error, but is more likely to follow a non-central Chi-square distribution (Browne 1984). This study investigates the empirical distributions of the ML and NWLS discrepancy functions. The study includes 13 different factor models with different types and degrees of specification error. It is found, except for small samples, that the empirical distribution of the ML-test statistic outperforms the empirical distribution of the NWLS-test statistic in terms of approximation to the theoretical non-central Chi-square distribution. Furthermore, in some cases, it turned out that the non-central Chi-square approximation was not appropriate even for models that contained minor and moderate degrees of specification error.

This paper studies the estimation and testing of general cointegrated systems by using an autoregressive approximation. Simple estimators for both the cointegration vectors and their weight matrix in the autoregressive error correction model representation of the system are developed. Since these estimators assume that the number of cointegration vectors and their normalization are fixed in advance, convenient specification tests for checking the validity of these assumptions are also provided. The asymptotic distributions of the estimators and test statistics are derived by assuming that the order of the auto-regressive approximation increases with the sample size at a suitable rate. This generalizes some previous results derived for finite-order autoregressions as no assumption of a finite-parameter data-generating process is imposed. The estimators and tests of the paper are interpreted in terms of autoregressive spectral density estimators at the zero frequency and, in the special case of a finite-order Gaussian autoregression, their relation to maximum likelihood procedures is discussed. All estimators of the paper can be applied with simple least-squares techniques and used to construct conventional Wald tests with asymptotic chi-square distributions under the null hypothesis. The limit theory of the specification tests is nonstandard, similar to that in univariate unit root tests.

A Shapiro-Wilk type goodness-of-fit test using a few order statistics

In this paper, we study penalized empirical likelihood for parameter estimation and variable selection in partially linear models with measurement errors in possibly all the variables. By using adaptive Lasso penalty function, we show that penalized empirical likelihood has the oracle property. That is, with probability tending to one, penalized empirical likelihood identifies the true model and estimates the nonzero coefficients as efficiently as if the sparsity of the true model was known in advance. Also, we introduce the penalized empirical likelihood ratio statistic to test a linear hypothesis of the parameter and prove that it follows an asymptotic Chi-square distribution under the null hypothesis. Some simulations and an application are given to illustrate the performance of the proposed method.

In this paper, we consider rank estimation for partial functional linear regression models based on functional principal component analysis. The proposed rank-based method is robust to outliers in the errors and highly efficient under a wide range of error distributions. The asymptotic properties of the resulting estimators are established under some regularity conditions. A simulation study conducted to investigate the finite sample performance of the proposed estimators shows that the proposed rank method performs well. Furthermore, the proposed methodology is illustrated by means of an analysis of the Berkeley growth data.

Recognition that measurement error in the independent (predictor) variable of data used for regression modeling necessitates special methods of model fitting dates back to the late 1800s. In the ensuing 100 years attention was devoted almost entirely to measurement error in linear regression models, at which time advances in computing opened the door to the nonlinear statistical modeling generally, and to the study of measurement error in nonlinear regression modeling in particular. This article contains an introduction to the problem of measurement error in regression using linear regression measurement error models to illustrate key ideas. It includes a discussion of the types of measurement error commonly studied (classical and Berkson, differential and nondifferential) and how those types arise in practice. A summary of the consequences of ignoring measurement error when analyzing data measured with error is presented, with detailed results given for simple and multiple linear measurement error models. Pointers to the literature on both linear and nonlinear measurement error models are provided.

The restricted minimum φ-divergence estimator, [Pardo, J.A., Pardo, L. and Zografos, K., 2002, Minimum φ-divergence estimators with constraints in multinomial populations. Journal of Statistical Planning and Inference, 104, 221–237], is employed to obtain estimates of the cell frequencies of an I×I contingency table under hypotheses of symmetry, marginal homogeneity or quasi-symmetry. The associated φ-divergence statistics are distributed asymptotically as chi-squared distributions under the null hypothesis. The new estimators and test statistics contain, as particular cases, the classical estimators and test statistics previously presented in the literature for the cited problems. A simulation study is presented, for the symmetry problem, to choose the best function φ2 for estimation and the best function φ1 for testing.

We propose a general framework for the specification testing of continuous treatment effect models. We assume a general residual function, which includes the average and quantile treatment effect models as special cases. The null models are identified under the unconfoundedness condition and contain a nonparametric weighting function. We propose a test statistic for the null model in which the weighting function is estimated by solving an expanding set of moment equations. We establish the asymptotic distributions of our test statistic under the null hypothesis and under fixed and local alternatives. The proposed test statistic is shown to be more efficient than that constructed from the true weighting function and can detect local alternatives deviated from the null models at the rate of . A simulation method is provided to approximate the null distribution of the test statistic. Monte-Carlo simulations show that our test exhibits a satisfactory finite-sample performance, and an application shows its practical value.