Semiparametric analysis of long-range dependence in nonlinear regression

This paper establishes consistency and asymptotic distribution theory for the least squares estimate of a vector parameter of non-linear regression with long-range dependent noise. A covariance-based estimate of the memory parameter is proposed. The consistency of the estimate is established.     

The Performance of a Robust Multistage Estimator in Nonlinear Regression with Heteroscedastic Errors

In this article, a robust multistage parameter estimator is proposed for nonlinear regression with heteroscedastic variance, where the residual variances are considered as a general parametric function of predictors. The motivation is based on considering the chi-square distribution for the calculated sample variance of the data. It is shown that outliers that are influential in nonlinear regression parameter estimates are not necessarily influential in calculating the sample variance. This matter persuades us, not only to robustify the estimate of the parameters of the models for both the regression function and the variance, but also to replace the sample variance of the data by a robust scale estimate.     

Sparse estimation and inference for censored median regression

Censored median regression has proved useful for analyzing survival data in complicated situations, say, when the variance is heteroscedastic or the data contain outliers. In this paper, we study the sparse estimation for censored median regression models, which is an important problem for high dimensional survival data analysis. In particular, a new procedure is proposed to minimize an inverse-censoring-probability weighted least absolute deviation loss subject to the adaptive LASSO penalty and result in a sparse and robust median estimator. We show that, with a proper choice of the tuning parameter, the procedure can identify the underlying sparse model consistently and has desired large-sample properties including root-n consistency and the asymptotic normality. The procedure also enjoys great advantages in computation, since its entire solution path can be obtained efficiently. Furthermore, we propose a resampling method to estimate the variance of the estimator. The performance of the procedure is illustrated by extensive simulations and two real data applications including one microarray gene expression survival data.     

Model robust confidence intervals

Confidence intervals are constructed for real-valued parameter estimation in a general regression model with normal errors. When the error variance is known these intervals are optimal (in the sense of minimizing length subject to guaranteed probability of coverage) among all intervals estimates which are centered at a linear estimate of the parameter. When the error variance is unknown and the regression model is an approximately linear model (a class of models which permits bounded systematic departures from an underlying ideal model) then an independent estimate of variance is found and the intervals can then be appropriately scaled.     

Selection between models through multi-step-ahead forecasting

We develop and show applications of two new test statistics for deciding if one ARIMA model provides significantly better h-step-ahead forecasts than another, as measured by the difference of approximations to their asymptotic mean square forecast errors. The two statistics differ in the variance estimates used for normalization. Both variance estimates are consistent even when the models considered are incorrect. Our main variance estimate is further distinguished by accounting for parameter estimation, while the simpler variance estimate treats parameters as fixed. Their broad consistency properties offer improvements to what are known as tests of Diebold and Mariano (1995) type, which are tests that treat parameters as fixed and use variance estimates that are generally not consistent in our context. We show how these statistics can be calculated for any pair of ARIMA models with the same differencing operator.     

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Summary. Searching for an effective dimension reduction space is an important problem in regression, especially for high dimensional data. We propose an adaptive approach based on semiparametric models, which we call the (conditional) minimum average variance estimation (MAVE) method, within quite a general setting. The MAVE method has the following advantages. Most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters (than for that of the nonparametric function). In contrast, a faster consistency rate can be achieved by the MAVE method even without undersmoothing the nonparametric link function estimator. The MAVE method is applicable to a wide range of models, with fewer restrictions on the distribution of the covariates, to the extent that even time series can be included. Because of the faster rate of consistency for the parameter estimators, it is possible for us to estimate the dimension of the space consistently. The relationship of the MAVE method with other methods is also investigated. In particular, a simple outer product gradient estimator is proposed as an initial estimator. In addition to theoretical results, we demonstrate the efficacy of the MAVE method for high dimensional data sets through simulation. Two real data sets are analysed by using the MAVE approach.     

Selection of bandwidth for kernel regression

ABSTRACT

The most important factor in kernel regression is a choice of a bandwidth. Considerable attention has been paid to extension the idea of an iterative method known for a kernel density estimate to kernel regression. Data-driven selectors of the bandwidth for kernel regression are considered. The proposed method is based on an optimally balanced relation between the integrated variance and the integrated square bias. This approach leads to an iterative quadratically convergent process. The analysis of statistical properties shows the rationale of the proposed method. In order to see statistical properties of this method the consistency is determined. The utility of the method is illustrated through a simulation study and real data applications.     

半参数纵向模型的惩罚二次推断函数估计

针对纵向数据半参数模型E(y|x,t)=XTβ+f(t),采用惩罚二次推断函数方法同时估计模型中的回归参数β和未知光滑函数f(t)。首先利用截断幂函数基对未知光滑函数进行基函数展开近似,然后利用惩罚样条的思想构造关于回归参数和基函数系数的惩罚二次推断函数,最小化惩罚二次推断函数便可得到回归参数和基函数系数的惩罚二次推断函数估计。理论结果显示,估计结果具有相合性和渐近正态性,通过数值方法也得到了较好的模拟结果。     

On the consistency of rank transformed regression

Rank-transformed regression (RTR) was proposed by Iman and Conover (1979) as an alternative to isotonic regression. This paper studies the consistency of the estimate obtained by RTR and show that, in general case, the estimate is not mean-square-error (MSE) consistent. The bias of the estimate is also studied by simulation.     

AN EXAMINATION OF ESTIMATED RESIDUALS IN A REGRESSION WITH AN INFINITE ORDER PARAMETRIC MODEL

We consider a linear regression with the error term that obeys an autoregressive model of infinite order and estimate parameters of the models. The parameters of the autoregressive model should be estimated based on estimated residuals obtained by means of the method of ordinary least squares, because the errors are unobservable. The consistency of the coefficients, variance and spectral density of the model obeyed by the error term is shown. Further, we estimate the coefficients of the linear regression by means of the method of estimated generalized least squares. We also show the consistency of the estimator.

     

On the performance of some new Liu parameters for the gamma regression model

The maximum likelihood (ML) method is used to estimate the unknown Gamma regression (GR) coefficients. In the presence of multicollinearity, the variance of the ML method becomes overstated and the inference based on the ML method may not be trustworthy. To combat multicollinearity, the Liu estimator has been used. In this estimator, estimation of the Liu parameter d is an important problem. A few estimation methods are available in the literature for estimating such a parameter. This study has considered some of these methods and also proposed some new methods for estimation of the d. The Monte Carlo simulation study has been conducted to assess the performance of the proposed methods where the mean squared error (MSE) is considered as a performance criterion. Based on the Monte Carlo simulation and application results, it is shown that the Liu estimator is always superior to the ML and recommendation about which best Liu parameter should be used in the Liu estimator for the GR model is given.     

Local CQR Smoothing: An Efficient and Safe Alternative to Local Polynomial Regression

Summary.  Local polynomial regression is a useful non-parametric regression tool to explore fine data structures and has been widely used in practice. We propose a new non-parametric regression technique called local composite quantile regression smoothing to improve local polynomial regression further. Sampling properties of the estimation procedure proposed are studied. We derive the asymptotic bias, variance and normality of the estimate proposed. The asymptotic relative efficiency of the estimate with respect to local polynomial regression is investigated. It is shown that the estimate can be much more efficient than the local polynomial regression estimate for various non-normal errors, while being almost as efficient as the local polynomial regression estimate for normal errors. Simulation is conducted to examine the performance of the estimates proposed. The simulation results are consistent with our theoretical findings. A real data example is used to illustrate the method proposed.     

Ratio test for variance change point in linear process with long memory

In this paper we consider the detection problem of variance change point in linear process with long memory. We propose the ratio test to detect the variance change point. The limiting distribution for test statistics under H ₀ is derived and the consistency of the test is also established. In comparison with the existing CUSUM of squares (SCUSUM) test, the ratio test does not need to estimate the long memory parameter in practical situation and therefore it can be used more conveniently.     

Choice of the ridge factor from the correlation matrix determinant

Ridge regression is the alternative method to ordinary least squares, which is mostly applied when a multiple linear regression model presents a worrying degree of collinearity. A relevant topic in ridge regression is the selection of the ridge parameter, and different proposals have been presented in the scientific literature. Since the ridge estimator is biased, its estimation is normally based on the calculation of the mean square error (MSE) without considering (to the best of our knowledge) whether the proposed value for the ridge parameter really mitigates the collinearity. With this goal and different simulations, this paper proposes to estimate the ridge parameter from the determinant of the matrix of correlation of the data, which verifies that the variance inflation factor (VIF) is lower than the traditionally established threshold. The possible relation between the VIF and the determinant of the matrix of correlation is also analysed. Finally, the contribution is illustrated with three real examples.     

Bayesian bridge regression

Classical bridge regression is known to possess many desirable statistical properties such as oracle, sparsity, and unbiasedness. One outstanding disadvantage of bridge regularization, however, is that it lacks a systematic approach to inference, reducing its flexibility in practical applications. In this study, we propose bridge regression from a Bayesian perspective. Unlike classical bridge regression that summarizes inference using a single point estimate, the proposed Bayesian method provides uncertainty estimates of the regression parameters, allowing coherent inference through the posterior distribution. Under a sparsity assumption on the high-dimensional parameter, we provide sufficient conditions for strong posterior consistency of the Bayesian bridge prior. On simulated datasets, we show that the proposed method performs well compared to several competing methods across a wide range of scenarios. Application to two real datasets further revealed that the proposed method performs as well as or better than published methods while offering the advantage of posterior inference.     

A note on smoothing parameter selection for penalized spline smoothing

In nonparametric regression the smoothing parameter can be selected by minimizing a Mean Squared Error (MSE) based criterion. For spline smoothing one can also rewrite the smooth estimation as a Linear Mixed Model where the smoothing parameter appears as the a priori variance of spline basis coefficients. This allows to employ Maximum Likelihood (ML) theory to estimate the smoothing parameter as variance component. In this paper the relation between the two approaches is illuminated for penalized spline smoothing (P-spline) as suggested in Eilers and Marx Statist. Sci. 11(2) (1996) 89. Theoretical and empirical arguments are given showing that the ML approach is biased towards undersmoothing, i.e. it chooses a too complex model compared to the MSE. The result is in line with classical spline smoothing, even though the asymptotic arguments are different. This is because in P-spline smoothing a finite dimensional basis is employed while in classical spline smoothing the basis grows with the sample size.     

Simulation-based consistent inference for biased working model of non-sparse high-dimensional linear regression

Variable selection in regression analysis is of importance because it can simplify model and enhance predictability. After variable selection, however, the resulting working model may be biased when it does not contain all of significant variables. As a result, the commonly used parameter estimation is either inconsistent or needs estimating high-dimensional nuisance parameter with very strong assumptions for consistency, and the corresponding confidence region is invalid when the bias is relatively large. We in this paper introduce a simulation-based procedure to reformulate a new model so as to reduce the bias of the working model, with no need to estimate high-dimensional nuisance parameter. The resulting estimators of the parameters in the working model are asymptotic normally distributed whether the bias is small or large. Furthermore, together with the empirical likelihood, we build simulation-based confidence regions for the parameters in the working model. The newly proposed estimators and confidence regions outperform existing ones in the sense of consistency.     

Detecting Variance Change-Points for Blocked Time Series and Dependent Panel Data

This article proposes a class of weighted differences of averages (WDA) statistics to test and estimate possible change-points in variance for time series with weakly dependent blocks and dependent panel data without specific distributional assumptions. We derive the asymptotic distributions of the test statistics for testing the existence of a single variance change-point under the null and local alternatives. We also study the consistency of the change-point estimator. Within the proposed class of the WDA test statistics, a standardized WDA test is shown to have the best consistency rate and is recommended for practical use. An iterative binary searching procedure is suggested for estimating the locations of possible multiple change-points in variance, whose consistency is also established. Simulation studies are conducted to compare detection power and number of wrong rejections of the proposed procedure to that of a cumulative sum (CUSUM) based test and a likelihood ratio-based test. Finally, we apply the proposed method to a stock index dataset and an unemployment rate dataset. Supplementary materials for this article are available online.     

Collinearity: revisiting the variance inflation factor in ridge regression

Ridge regression has been widely applied to estimate under collinearity by defining a class of estimators that are dependent on the parameter k. The variance inflation factor (VIF) is applied to detect the presence of collinearity and also as an objective method to obtain the value of k in ridge regression. Contrarily to the definition of the VIF, the expressions traditionally applied in ridge regression do not necessarily lead to values of VIFs equal to or greater than 1. This work presents an alternative expression to calculate the VIF in ridge regression that satisfies the aforementioned condition and also presents other interesting properties.