Semiparametric Regression Models with Applications to Scoring: A Review

Semiparametric regression models have been proposed in the econometric literature as a trade-off between the simple but easily implementable and interpretable parametric models and the flexible but structure free smoothing techniques. Some semiparametric models for binary response with possible application to scoring data are reviewed: single-index models, generalized partially linear models, generalized partially linear single-index models, and multiple-index models. All these models are extensions of the classical logistic regression.     

Nonparametric Estimation of Average Growth Curve with General Nonstationary Error Process

The nonparametric estimation of the growth curve has been extensively studied in both stationary and some nonstationary particular situations. In this work, we consider the statistical problem of estimating the average growth curve for a fixed design model with nonstationary error process. The nonstationarity considered here is of a general form, and this article may be considered as an extension of previous results. The optimal bandwidth is shown to depend on the singularity of the autocovariance function of the error process along the diagonal. A Monte Carlo study is conducted in order to assess the influence of the number of subjects and the number of observations per subject on the estimation.     

Using regression mixture models with non-normal data: examining an ordered polytomous approach

Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; 3000 observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the 10 scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two-class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.     

A Simple Estimator of Error Correlation in Non-parametric Regression Models

Abstract.  It is well known that major strength of non-parametric regression function estimation breaks down when correlated errors exist in the data. Positively (negatively) correlated errors tend to produce undersmoothing (oversmoothing). Several remedies have been proposed in the context of bandwidth selection problem, but they are hard to implement without prior knowledge of error correlations. In this paper we propose a simple estimator of error correlation which is ready to implement and reports a reasonably good performance.     

Uniform Test for Predictive Regression With AR Errors

Testing predictability is of importance in economics and finance. Based on a predictive regression model with independent and identically distributed errors, some uniform tests have been proposed in the literature without distinguishing whether the predicting variable is stationary or nearly integrated. In this article, we extend the empirical likelihood methods of Zhu, Cai, and Peng with independent errors to the case of an AR error process. Again, the proposed new tests do not need to know whether the predicting variable is stationary or nearly integrated, and whether it has a finite variance or an infinite variance. A simulation study shows the new methodologies perform well in finite sample.     

Reml estimation for repeated measures analysis

Models for repeated measures or growth curves consist of a mean response plus error and the errors are usually correlated. Both maximum likelihood and residual maximum likelihood (REML) estimators of a regression model with dependent errors are derived for cases in which the variance matrix of the error model admits a convenient Cholesky factorisation. This factorisation may be linked to methods for producing recursive estimates of the regression parameters and recursive residuals to provide a convenient computational method. The method is used to develop a general approach to repeated measures analysis.     

信用评分模型中拒绝推断问题研究：基于半监督协同训练法的改进

随着我国金融市场的蓬勃发展,信用评价中的拒绝推断问题越来越受到重视。针对信用评分模型中存在的有类别标签的样本占比低,并且样本中的类别分布不平衡等问题,本文在半监督学习技术与集成学习理论的基础上,提出了一种新的算法——BCT算法。该算法通过使用动态Bagging生成多个子分类器,引入分类阈值参数来解决样本类别分布不平衡问题,以及设定早停止条件来避免算法迭代过程中存在的过拟合风险,以此对传统半监督协同训练法进行改进。通过在5个真实数据集上的实证分析发现,在不同数据集与不同拒绝比例下,BCT算法的性能均优于其他6种有监督学习和半监督学习算法的信用评分模型,显示了BCT算法具有良好的模型泛化性能和更高的模型评价能力。     

Nonparametric Kernel Methods with Errors‐in‐Variables: Constructing Estimators,Computing them,and Avoiding Common Mistakes

Estimating a curve nonparametrically from data measured with error is a difficult problem that has been studied by many authors. Constructing a consistent estimator in this context can sometimes be quite challenging, and in this paper we review some of the tools that have been developed in the literature for kernel‐based approaches, founded on the Fourier transform and a more general unbiased score technique. We use those tools to rederive some of the existing nonparametric density and regression estimators for data contaminated by classical or Berkson errors, and discuss how to compute these estimators in practice. We also review some mistakes made by those working in the area, and highlight a number of problems with an existing R package decon .     

Modeling a cross-sectional response variable with longitudinal predictors: an example of pulse pressure and pulse wave velocity

We wish to model pulse wave velocity (PWV) as a function of longitudinal measurements of pulse pressure (PP) at the same and prior visits at which the PWV is measured. A number of approaches are compared. First, we use the PP at the same visit as the PWV in a linear regression model. In addition, we use the average of all available PPs as the explanatory variable in a linear regression model. Next, a two-stage process is applied. The longitudinal PP is modeled using a linear mixed-effects model. This modeled PP is used in the regression model to describe PWV. An approach for using the longitudinal PP data is to obtain a measure of the cumulative burden, the area under the PP curve. This area under the curve is used as an explanatory variable to model PWV. Finally, a joint Bayesian model is constructed similar to the two-stage model.     

Performance of Wald-type estimator for parametric component in partial linear regression with a mixture of Berkson and classical error models

This article discusses a consistent and almost unbiased estimation approach in partial linear regression for parameters of interest when the regressors are contaminated with a mixture of Berkson and classical errors. Advantages of the presented procedure are: (1) random errors and observations are not necessarily to be parametric settings; (2) there is no need to use additional sample information, and to consider the estimation of nuisance parameters. We will examine the performance of our presented estimate in a variety of numerical examples through Monte Carlo simulation. The proposed approach is also illustrated in the analysis of an air pollution data.     

Locally Efficient Semiparametric Estimators for Proportional Hazards Models with Measurement Error

We propose a new class of semiparametric estimators for proportional hazards models in the presence of measurement error in the covariates, where the baseline hazard function, the hazard function for the censoring time, and the distribution of the true covariates are considered as unknown infinite dimensional parameters. We estimate the model components by solving estimating equations based on the semiparametric efficient scores under a sequence of restricted models where the logarithm of the hazard functions are approximated by reduced rank regression splines. The proposed estimators are locally efficient in the sense that the estimators are semiparametrically efficient if the distribution of the error‐prone covariates is specified correctly and are still consistent and asymptotically normal if the distribution is misspecified. Our simulation studies show that the proposed estimators have smaller biases and variances than competing methods. We further illustrate the new method with a real application in an HIV clinical trial.     

Change-Point Tests for the Error Distribution in Non-parametric Regression

Abstract.  Several testing procedures are proposed that can detect change-points in the error distribution of non-parametric regression models. Different settings are considered where the change-point either occurs at some time point or at some value of the covariate. Fixed as well as random covariates are considered. Weak convergence of the suggested difference of sequential empirical processes based on non-parametrically estimated residuals to a Gaussian process is proved under the null hypothesis of no change-point. In the case of testing for a change in the error distribution that occurs with increasing time in a model with random covariates the test statistic is asymptotically distribution free and the asymptotic quantiles can be used for the test. This special test statistic can also detect a change in the regression function. In all other cases the asymptotic distribution depends on unknown features of the data-generating process and a bootstrap procedure is proposed in these cases. The small sample performances of the proposed tests are investigated by means of a simulation study and the tests are applied to a data example.     

A robust and efficient estimation method for nonparametric models with jump points

Nonparametric models with jump points have been considered by many researchers. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. In this article, a local piecewise-modal method is proposed to estimate the regression function with jump points in nonparametric models, and a piecewise-modal EM algorithm is introduced to estimate the proposed estimator. Under some regular conditions, the large-sample theory is established for the proposed estimators. Several simulations are presented to evaluate the performances of the proposed method, which shows that the proposed estimator is more efficient than the local piecewise-polynomial regression estimator in the presence of outliers or heavy tail error distribution. What is more, the proposed procedure is asymptotically equivalent to the local piecewise-polynomial regression estimator under the assumption that the error distribution is a Gaussian distribution. The proposed method is further illustrated via the sea-level pressures.     

Minimum distance estimation in linear regression with strong mixing errors

Abstract

Minimum distance estimation on the linear regression model with independent errors is known to yield an efficient and robust estimator. We extend the method to the model with strong mixing errors and obtain an estimator of the vector of the regression parameters. The goal of this article is to demonstrate the proposed estimator still retains efficiency and robustness. To that end, this article investigates asymptotic distributional properties of the proposed estimator and compares it with other estimators. The efficiency and the robustness of the proposed estimator are empirically shown, and its superiority over the other estimators is established.     

Estimation of Slope for Linear Regression Model with Uncertain Prior Information and Student-t Error

This article considers estimation of the slope parameter of the linear regression model with Student-t errors in the presence of uncertain prior information on the value of the unknown slope. Incorporating uncertain non sample prior information with the sample data the unrestricted, restricted, preliminary test, and shrinkage estimators are defined. The performances of the estimators are compared based on the criteria of unbiasedness and mean squared errors. Both analytical and graphical methods are explored. Although none of the estimators is uniformly superior to the others, if the non sample information is close to its true value, the shrinkage estimator over performs the rest of the estimators.     

Joint detection for functional polynomial regression with autoregressive errors

In this article, we are concerned with detecting the true structure of a functional polynomial regression with autoregressive (AR) errors. The first issue is to detect which orders of the polynomial are significant in functional polynomial regression. The second issue is to detect which orders of the AR process in the AR errors are significant. We propose a shrinkage method to deal with the two problems: polynomial order selection and autoregressive order selection. Simulation studies demonstrate that the new method can identify the true structure. One empirical example is also presented to illustrate the usefulness of our method.     

High-order data sharpening with dependent errors for regression bias reduction

Abstract

In this paper, we show that Y can be introduced into data sharpening to produce non-parametric regression estimators that enjoy high orders of bias reduction. Compared with those in existing literature, the proposed data-sharpening estimator has advantages including simplicity of the estimators, good performance of expectation and variance, and mild assumptions. We generalize this estimator to dependent errors. Finally, we conduct a limited simulation to illustrate that the proposed estimator performs better than existing ones.     

The Performance of Robust Two-Stage Estimator in Nonlinear Regression with Autocorrelated Error

Some statistics practitioners often ignore the underlying assumptions when analyzing a real data and employ the Nonlinear Least Squares (NLLS) method to estimate the parameters of a nonlinear model. In order to make reliable inferences about the parameters of a model, require that the underlying assumptions, especially the assumption that the errors are independent, are satisfied. However, in a real situation, we may encounter dependent error terms which prone to produce autocorrelated errors. A two-stage estimator (CTS) has been developed to remedy this problem. Nevertheless, it is now evident that the presence of outliers have an unduly effect on the least squares estimates. We expect that the CTS is also easily affected by outliers since it is based on the least squares estimator, which is not robust. In this article, we propose a Robust Two-Stage (RTS) procedure for the estimation of the nonlinear regression parameters in the situation where autocorrelated errors come together with the existence of outliers. The numerical example and simulation study signify that the RTS is more efficient than the NLLS and the CTS methods.     

Quasi-Minimax Estimation in the Linear Model with Measurement Errors

In this article, we consider quasi-minimax estimation in the linear regression model where some covariates are measured with additive errors. When measurement errors are directly ignored the minimax risk of the resulting estimator can be large. By correcting the attenuation we propose a penalized quadratic risk function. A simulation study is conducted to illustrate the performance of the proposed estimators.     

Statistical inferences for single-index models with measurement errors

An important factor in house prices is its location. However, measurement errors arise frequently in the process of observing variables such as the latitude and longitude of the house. The single-index models with measurement errors are used to study the relationship between house location and house price. We obtain the estimators by a SIMEX method based on the local linear method and the estimating equation. To test the significance of the index coefficient and the linearity of the link function, we establish the generalized likelihood ratio (GLR) tests for the models. We demonstrate that the asymptotic null distributions of the established GLR tests follow χ2-distributions which are independent of nuisance parameters or functions. Finally, two simulated examples and a real estate valuation data set are given to illustrate the effect of GLR tests.