Testing the Significance of Categorical Predictor Variables in Nonparametric Regression Models

In this paper we propose a test for the significance of categorical predictors in nonparametric regression models. The test is fully data-driven and employs cross-validated smoothing parameter selection while the null distribution of the test is obtained via bootstrapping. The proposed approach allows applied researchers to test hypotheses concerning categorical variables in a fully nonparametric and robust framework, thereby deflecting potential criticism that a particular finding is driven by an arbitrary parametric specification. Simulations reveal that the test performs well, having significantly better power than a conventional frequency-based nonparametric test. The test is applied to determine whether OECD and non-OECD countries follow the same growth rate model or not. Our test suggests that OECD and non-OECD countries follow different growth rate models, while the tests based on a popular parametric specification and the conventional frequency-based nonparametric estimation method fail to detect any significant difference.     

Finding local departures from a parametric model using nonparametric regression

Goodness-of-fit evaluation of a parametric regression model is often done through hypothesis testing, where the fit of the model of interest is compared statistically to that obtained under a broader class of models. Nonparametric regression models are frequently used as the latter type of model, because of their flexibility and wide applicability. To date, this type of tests has generally been performed globally, by comparing the parametric and nonparametric fits over the whole range of the data. However, in some instances it might be of interest to test for deviations from the parametric model that are localized to a subset of the data. In this case, a global test will have low power and hence can miss important local deviations. Alternatively, a naive testing approach that discards all observations outside the local interval will suffer from reduced sample size and potential overfitting. We therefore propose a new local goodness-of-fit test for parametric regression models that can be applied to a subset of the data but relies on global model fits, and propose a bootstrap-based approach for obtaining the distribution of the test statistic. We compare the new approach with the global and the naive tests, both theoretically and through simulations, and illustrate its practical behavior in an application. We find that the local test has a better ability to detect local deviations than the other two tests.     

Outlier Resistant Estimation in Difference-Based Semiparametric Partially Linear Models

Partially linear models are extensions of linear models that include a nonparametric function of some covariate allowing an adequate and more flexible handling of explanatory variables than in linear models. The difference-based estimation in partially linear models is an approach designed to estimate parametric component by using the ordinary least squares estimator after removing the nonparametric component from the model by differencing. However, it is known that least squares estimates do not provide useful information for the majority of data when the error distribution is not normal, particularly when the errors are heavy-tailed and when outliers are present in the dataset. This paper aims to find an outlier-resistant fit that represents the information in the majority of the data by robustly estimating the parametric and the nonparametric components of the partially linear model. Simulations and a real data example are used to illustrate the feasibility of the proposed methodology and to compare it with the classical difference-based estimator when outliers exist.     

A consistent test for heteroscedasticity in semi-parametric regression with nonparametric variance function based on the kernel method

It is important to detect the variance heterogeneity in regression models. Heteroscedasticity tests have been well studied in parametric and nonparametric regression models. This paper presents a consistent test for heteroscedasticity for nonlinear semi-parametric regression models with nonparametric variance function based on the kernel method. The properties of the test are investigated through Monte Carlo simulations. The test methods are illustrated with a real example.     

Restricted Estimation in Partially Linear Errors-in-Variables Models

As a compromise between parametric regression and nonparametric regression, partially linear models are frequently used in statistical modelling. This article considers statistical inference for this semiparametric model when the linear covariate is measured with additive error and some additional linear restrictions on the parametric component are assumed to hold. We propose a restricted corrected profile least-squares estimator for the parametric component, and study the asymptotic normality of the estimator. To test hypothesis on the parametric component, we construct a Wald test statistic and obtain its limiting distribution. Some simulation studies are conducted to illustrate our approaches.     

Direct parametric inference for the cumulative incidence function

Summary.  In survival data that are collected from phase III clinical trials on breast cancer, a patient may experience more than one event, including recurrence of the original cancer, new primary cancer and death. Radiation oncologists are often interested in comparing patterns of local or regional recurrences alone as first events to identify a subgroup of patients who need to be treated by radiation therapy after surgery. The cumulative incidence function provides estimates of the cumulative probability of locoregional recurrences in the presence of other competing events. A simple version of the Gompertz distribution is proposed to parameterize the cumulative incidence function directly. The model interpretation for the cumulative incidence function is more natural than it is with the usual cause-specific hazard parameterization. Maximum likelihood analysis is used to estimate simultaneously parametric models for cumulative incidence functions of all causes. The parametric cumulative incidence approach is applied to a data set from the National Surgical Adjuvant Breast and Bowel Project and compared with analyses that are based on parametric cause-specific hazard models and nonparametric cumulative incidence estimation.     

基于金融高频数据的ACD模型非参数设定检验

 目前关于ACD的实证研究已经十分丰富，却很少有人把注意力放在ACD及其扩展模型设定的检验上，本文采用的D检验就是通过衡量残差密度函数的参数和非参数估计值之间的紧密程度，来检验模型设定的优劣。     

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.     

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

The joint modeling of longitudinal and survival data has received extraordinary attention in the statistics literature recently, with models and methods becoming increasingly more complex. Most of these approaches pair a proportional hazards survival with longitudinal trajectory modeling through parametric or nonparametric specifications. In this paper we closely examine one data set previously analyzed using a two parameter parametric model for Mediterranean fruit fly (medfly) egg-laying trajectories paired with accelerated failure time and proportional hazards survival models. We consider parametric and nonparametric versions of these two models, as well as a proportional odds rate model paired with a wide variety of longitudinal trajectory assumptions reflecting the types of analyses seen in the literature. In addition to developing novel nonparametric Bayesian methods for joint models, we emphasize the importance of model selection from among joint and non joint models. The default in the literature is to omit at the outset non joint models from consideration. For the medfly data, a predictive diagnostic criterion suggests that both the choice of survival model and longitudinal assumptions can grossly affect model adequacy and prediction. Specifically for these data, the simple joint model used in by Tseng et al. (Biometrika 92:587–603, 2005) and models with much more flexibility in their longitudinal components are predictively outperformed by simpler analyses. This case study underscores the need for data analysts to compare on the basis of predictive performance different joint models and to include non joint models in the pool of candidates under consideration.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

A unified approach to estimation of nonlinear mixed effects and Berkson measurement error models

Mixed effects models and Berkson measurement error models are widely used. They share features which the author uses to develop a unified estimation framework. He deals with models in which the random effects (or measurement errors) have a general parametric distribution, whereas the random regression coefficients (or unobserved predictor variables) and error terms have nonparametric distributions. He proposes a second-order least squares estimator and a simulation-based estimator based on the first two moments of the conditional response variable given the observed covariates. He shows that both estimators are consistent and asymptotically normally distributed under fairly general conditions. The author also reports Monte Carlo simulation studies showing that the proposed estimators perform satisfactorily for relatively small sample sizes. Compared to the likelihood approach, the proposed methods are computationally feasible and do not rely on the normality assumption for random effects or other variables in the model.     

A class of nonparametric bivariate survival function estimators for randomly censored and truncated data

This paper proposes a class of nonparametric estimators for the bivariate survival function estimation under both random truncation and random censoring. In practice, the pair of random variables under consideration may have certain parametric relationship. The proposed class of nonparametric estimators uses such parametric information via a data transformation approach and thus provides more accurate estimates than existing methods without using such information. The large sample properties of the new class of estimators and a general guidance of how to find a good data transformation are given. The proposed method is also justified via a simulation study and an application on an economic data set.     

Estimation and inference for generalized semi-varying coefficient models

This paper is concerned with the estimation and inference in generalized semi-varying coefficient models. An orthogonal projection local quasi-likelihood estimation is investigated, which can easily be used to estimate the model parametric and nonparametric parts. Then an empirical likelihood logarithmic approach to construct the confidence regions/intervals of the nonparametric parts is developed. Under some mild conditions, the asymptotic properties of the resulting estimators are studied explicitly, respectively. Some simulation studies are carried out to examine the finite sample performance of the proposed methods. Finally, the methodologies are illustrated by a real data set.     

A new algorithm to estimate monotone nonparametric link functions and a comparison with parametric approach

The generalized linear model (GLM) is a class of regression models where the means of the response variables and the linear predictors are joined through a link function. Standard GLM assumes the link function is fixed, and one can form more flexible GLM by either estimating the flexible link function from a parametric family of link functions or estimating it nonparametically. In this paper, we propose a new algorithm that uses P-spline for nonparametrically estimating the link function which is guaranteed to be monotone. It is equivalent to fit the generalized single index model with monotonicity constraint. We also conduct extensive simulation studies to compare our nonparametric approach for estimating link function with various parametric approaches, including traditional logit, probit and robit link functions, and two recently developed link functions, the generalized extreme value link and the symmetric power logit link. The simulation study shows that the link function estimated nonparametrically by our proposed algorithm performs well under a wide range of different true link functions and outperforms parametric approaches when they are misspecified. A real data example is used to illustrate the results.     

Prediction bands for the EDF of a future sample

We develop both nonparametric and parametric methods for obtaining prediction bands for the empirical distribution function (EDF) of a future sample. These methods yield simultaneous prediction intervals for all order statistics of the future sample, and they also correspond to tests for the two-sample problem. The nonparametric prediction bands correspond to the two-sample Kolmogorov-Smirnov test and related nonparametric tests, but the parametric prediction bands correspond to entirely new parametric two-sample tests. The parametric prediction bands tend to outperform the nonparametric bands when the parametric assumptions hold, but they may have true coverage probabilities well below their nominal levels when the parametric assumptions fail. A new computational algorithm is used to obtain critical values in the nonparametric case.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

Local maximum likelihood estimation and inference

Local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique. It extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces. Associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment. This paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation. The approach is then applied to least squares nonparametric regression and to nonparametric logistic regression. Our experiences in these two settings show that the general idea outlined here is powerful and encouraging.     

Partially Linear Additive Models with Unknown Link Functions

In this paper, we consider partially linear additive models with an unknown link function, which include single‐index models and additive models as special cases. We use polynomial spline method for estimating the unknown link function as well as the component functions in the additive part. We establish that convergence rates for all nonparametric functions are the same as in one‐dimensional nonparametric regression. For a faster rate of the parametric part, we need to define appropriate ‘projection’ that is more complicated than that defined previously for partially linear additive models. Compared to previous approaches, a distinct advantage of our estimation approach in implementation is that estimation directly reduces estimation in the single‐index model and can thus deal with much larger dimensional problems than previous approaches for additive models with unknown link functions. Simulations and a real dataset are used to illustrate the proposed model.     

Nonparametric estimation of a switching regression model

High-dimensional data often exhibit multi-collinearity, leading to unstable regression coefficients. To address sample selection bias and problems associated with high dimensionality, principal components were extracted and used as predictors in a switching regression model. Since principal component regression often results to decline in predictive ability due to the selection of few principal components, we formulate the model with nonparametric function of principal components in lieu of individual predictors. Simulation studies indicated better predictive ability for nonparametric principal component switching regression over the parametric counterpart while mitigating the adverse effects of multi-collinearity and high dimensionality.     

A new parametric family for modelling cumulative incidence functions: application to breast cancer data

Summary.  Competing risks situations can be encountered in many research areas such as medicine, social science and engineering. The main stream of analyses of those competing risks data has been nonparametric or semiparametric in the statistical literature. We propose a new parametric family to parameterize the cumulative incidence function completely. The new distribution is sufficiently flexible to fit various shapes of hazard patterns in survival data and increases the efficiency of the cumulative incidence estimates over the distribution-free approaches. A simple two-sample parametric test statistic is also proposed to compare the cumulative incidence functions between two groups at a given time point. The new parametric approach is illustrated by using breast cancer data sets from the National Surgical Adjuvant Breast and Bowel Project.