Minimum hellinger distance estimation for normal models

A robust estimator introduced by Beran (1977a, 1977b), which is based on the minimum Hellinger distance between a projection model density and a nonparametric sample density, is studied empirically. An extensive simulation provides an estimate of the small sample distribution and supplies empirical evidence of the estimator performance for a normal location-scale model. While the performance of the minimum Hellinger distance estimator is seen to be competitive with the maximum likelihood estimator at the true model, its robustness to deviations from normality is shown to be competitive in this setting with that obtained from the M-estimator and the Cramér-von Mises minimum distance estimator. Beran also introduced a goodness-of-fit statisticH ₂, based on the minimized Hellinger distance between a member of a parametric family of densities and a nonparametric density estimate. We investigate the statistic H (the square root of H ₂) as a test for normality when both location and scale are unspecified. Empirically derived critical values are given which do not require extensive tables. The power of the statistic H compares favorably with four other widely used tests for normality.     

Automatic bandwidth selection for modified m-smoother ∗

Traditional parametric and nonparametric regression techniques encounter serious over smoothing problems when jump point discontinuities exist in the underlying mean function. Recently, Chu, Glad, Godtliebsen and Marron (1998) developed a method using a modified M-smoothing technique to preserve jumps and spikes while producing a smooth estimate of the mean function. The performance of Chu etal.'s (1998) method is quite sensitive to the choice of the required bandwidths g and h. Furthermore, it is not obvious how to extend certain commonly used automatic bandwidth selection procedures when jumps and spikes are present. In this paper we propose a rule of thumb method of choosing the smoothing parameters based on asymptotic optimal bandwidth formulas and robust estimates of unknown quantities. We also evaluate the proposed bandwidth selection method via a small simulation study.     

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

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.     

Nonparametric Tests for Comparing Several Coefficients of Variation

We consider the problem of comparing (k + 1) coefficients of variation. We are interested in testing the null hypothesis that the coefficients of variation are equal against each of the alternatives: (a) some populations have different coefficients of variation and (b) the coefficients of variation are ordered. Three nonparametric test statistics are proposed and their asymptotic theory is developed. We compared the proposed tests together with another parametric test using two Monte Carlo studies to estimate their probabilities of Type I error and powers. An illustration of the proposed tests using a real data set is given.     

Statistical inference for a single-index varying-coefficient model

We investigate the estimators of parameters of interest for a single-index varying-coefficient model. To estimate the unknown parameter efficiently, we first estimate the nonparametric component using local linear smoothing, then construct an estimator of parametric component by using estimating equations. Our estimator for the parametric component is asymptotically efficient, and the estimator of nonparametric component has asymptotic normality and optimal uniform convergence rate. Our results provide ways to construct confidence regions for the involved unknown parameters. The finite-sample behavior of the new estimators is evaluated through simulation studies, and applications to two real data are illustrated.     

An Asymptotic Characterization of Finite Degree U-statistics With Sample Size-Dependent Kernels: Applications to Nonparametric Estimators and Test Statistics

We provide a simple result on the H-decomposition of a U-statistics that allows for easy determination of its magnitude when the statistic’s kernel depends on the sample size n. The result provides a direct and convenient method to characterize the asymptotic magnitude of semiparametric and nonparametric estimators or test statistics involving high dimensional sums. We illustrate the use of our result in previously studied estimators/test statistics and in a novel nonparametric R² test for overall significance of a nonparametric regression model.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

Asymptotic properties of mean cumulative function estimators from window-observation recurrence data

A variety of nonparametric and parametric methods have been used to estimate the mean cumulative function (MCF) for the recurrence data collected from the counting process. When the recurrence histories of some units are available in disconnected observation windows with gaps in between, Zuo et al. (2008) showed that both the nonparametric and parametric methods can be extended to estimate the MCF. In this article, we establish some asymptotic properties of the MCF estimators for the window-observation recurrence data.     

A note on the multivariate multisample rank sum and median tests

The muitivariate nonparametric tests analogous to the univar-iate rank sum test and median test are contained in Puri and Sen (1970). These tests provided a practical alternative for the analysis of multivariate data when the assumptions of parametric methods are not satisfied.

In this paper maximum values for L_Nthe asymptotic chi-Square test statistic for both the Multivariate Multisample Rank Sum Test (MMRST) and the Multivariate Multisample Median Test (MMMT) are developed.     

Generalized estimating equations for mixtures with varying concentrations

A finite mixture model is considered in which the mixing probabilities vary from observation to observation. A parametric model is assumed for one mixture component distribution, while the others are nonparametric nuisance parameters. Generalized estimating equations (GEE) are proposed for the semi‐parametric estimation. Asymptotic normality of the GEE estimates is demonstrated and the lower bound for their dispersion (asymptotic covariance) matrix is derived. An adaptive technique is developed to derive estimates with nearly optimal small dispersion. An application to the sociological analysis of voting results is discussed. The Canadian Journal of Statistics 41: 217–236; 2013 © 2013 Statistical Society of Canada     

A semiparametric approach for modelling multivariate nonlinear time series

In this article, a semiparametric time‐varying nonlinear vector autoregressive (NVAR) model is proposed to model nonlinear vector time series data. We consider a combination of parametric and nonparametric estimation approaches to estimate the NVAR function for both independent and dependent errors. We use the multivariate Taylor series expansion of the link function up to the second order which has a parametric framework as a representation of the nonlinear vector regression function. After the unknown parameters are estimated by the maximum likelihood estimation procedure, the obtained NVAR function is adjusted by a nonparametric diagonal matrix, where the proposed adjusted matrix is estimated by the nonparametric kernel estimator. The asymptotic consistency properties of the proposed estimators are established. Simulation studies are conducted to evaluate the performance of the proposed semiparametric method. A real data example on short‐run interest rates and long‐run interest rates of United States Treasury securities is analyzed to demonstrate the application of the proposed approach. The Canadian Journal of Statistics 47: 668–687; 2019 © 2019 Statistical Society of Canada     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

On Semiparametric Mode Regression Estimation

It has been found that, for a variety of probability distributions, there is a surprising linear relation between mode, mean, and median. In this article, the relation between mode, mean, and median regression functions is assumed to follow a simple parametric model. We propose a semiparametric conditional mode (mode regression) estimation for an unknown (unimodal) conditional distribution function in the context of regression model, so that any m-step-ahead mean and median forecasts can then be substituted into the resultant model to deliver m-step-ahead mode prediction. In the semiparametric model, Least Squared Estimator (LSEs) for the model parameters and the simultaneous estimation of the unknown mean and median regression functions by the local linear kernel method are combined to infer about the parametric and nonparametric components of the proposed model. The asymptotic normality of these estimators is derived, and the asymptotic distribution of the parameter estimates is also given and is shown to follow usual parametric rates in spite of the presence of the nonparametric component in the model. These results are applied to obtain a data-based test for the dependence of mode regression over mean and median regression under a regression model.     

Estimation of regression parameters in a semiparametric transformation model

A semiparametric approach to model skewed/heteroscedastic regression data is discussed. We work with a semiparametric transform-both-sides regression model, which contains a parametric regression function and a nonparametric transformation. This model is adequate when the relationship between the median response and the explanatory variable has been specified by a theoretical result or a previous empirical study. The transform-both-sides model with a parametric transformation has been studied extensively and applied successfully to a number data sets. Allowing a nonparametric transformation function increases the flexibility of the model. In this article, we estimate the nonparametric transformation function by the conditional kernel density approach developed by Wang and Ruppert (1995), and then use a pseudo-maximum likelihood estimator to estimate the regression parameters. This estimate of the regression parameters has not been studied previously. In this article, the asymptotic distribution of this pseudo-MLE is derived. We also show that when σ, the standard deviation of the error, goes to zero (small σ asymptotics), this estimator is adaptive. Adaptive means that the regression parameters are estimated as precisely as when the transformation is known exactly. A similar result holds in the parametric approaches of Carroll and Ruppert (1984) and Ruppert and Aldershof (1989). Simulated and real examples are provided to illustrate the performance of the proposed estimator for finite sample size.     

Comparison of Nonparametric Components in Two Partially Linear Models

In this article, we are concerned with whether the nonparametric functions are parallel from two partial linear models, and propose a test statistic to check the difference of the two functions. The unknown constant α is estimated by using moment method under null models. Nonparametric functions under both null and full models are estimated by using local linear method. The asymptotic properties of parametric and nonparametric components are derived. The test statistic under the null hypothesis is calculated and shown to be asymptotically normal.     

Nonparametric Covariate-Adjusted Association Tests Based on the Generalized Kendall's Tau()

Identifying the risk factors for comorbidity is important in psychiatric research. Empirically, studies have shown that testing multiple, correlated traits simultaneously is more powerful than testing a single trait at a time in association analysis. Furthermore, for complex diseases, especially mental illnesses and behavioral disorders, the traits are often recorded in different scales such as dichotomous, ordinal and quantitative. In the absence of covariates, nonparametric association tests have been developed for multiple complex traits to study comorbidity. However, genetic studies generally contain measurements of some covariates that may affect the relationship between the risk factors of major interest (such as genes) and the outcomes. While it is relatively easy to adjust these covariates in a parametric model for quantitative traits, it is challenging for multiple complex traits with possibly different scales. In this article, we propose a nonparametric test for multiple complex traits that can adjust for covariate effects. The test aims to achieve an optimal scheme of adjustment by using a maximum statistic calculated from multiple adjusted test statistics. We derive the asymptotic null distribution of the maximum test statistic, and also propose a resampling approach, both of which can be used to assess the significance of our test. Simulations are conducted to compare the type I error and power of the nonparametric adjusted test to the unadjusted test and other existing adjusted tests. The empirical results suggest that our proposed test increases the power through adjustment for covariates when there exist environmental effects, and is more robust to model misspecifications than some existing parametric adjusted tests. We further demonstrate the advantage of our test by analyzing a data set on genetics of alcoholism.     

Strictly monotone and smooth nonparametric regression for two or more variables

The authors propose a new monotone nonparametric estimate for a regression function of two or more variables. Their method consists in applying successively one‐dimensional isotonization procedures on an initial, unconstrained nonparametric regression estimate. In the case of a strictly monotone regression function, they show that the new estimate and the initial one are first‐order asymptotic equivalent; they also establish asymptotic normality of an appropriate standardization of the new estimate. In addition, they show that if the regression function is not monotone in one of its arguments, the new estimate and the initial one have approximately the same L^p‐norm. They illustrate their approach by means of a simulation study, and two data examples are analyzed.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

Asymptotic properties of the GMLE in the case 1 interval-censorship model with discrete inspection times

We consider the case 1 interval censorship model in which the survival time has an arbitrary distribution function F₀ and the inspection time has a discrete distribution function G. In such a model one is only able to observe the inspection time and whether the value of the survival time lies before or after the inspection time. We prove the strong consistency of the generalized maximum-likelihood estimate (GMLE) of the distribution function F₀ at the support points of G and its asymptotic normality and efficiency at what we call regular points. We also present a consistent estimate of the asymptotic variance at these points. The first result implies uniform strong consistency on [0, ∞) if F₀ is continuous and the support of G is dense in [0, ∞). For arbitrary F₀ and G, Peto (1973) and Tumbull (1976) conjectured that the convergence for the GMLE is at the usual parametric rate n½ Our asymptotic normality result supports their conjecture under our assumptions. But their conjecture was disproved by Groeneboom and Wellner (1992), who obtained the nonparametric rate ni under smoothness assumptions on the F₀ and G.