Nonlinear Censored Regression Using Synthetic Data

Abstract.  The problem of estimating a nonlinear regression model, when the dependent variable is randomly censored, is considered. The parameter of the model is estimated by least squares using synthetic data. Consistency and asymptotic normality of the least squares estimators are derived. The proofs are based on a novel approach that uses i.i.d. representations of synthetic data through Kaplan–Meier integrals. The asymptotic results are supported by a small simulation study.     

ERROR VARIANCE ESTIMATION FOR THE SINGLE‐INDEX MODEL

Single‐index models provide one way of reducing the dimension in regression analysis. The statistical literature has focused mainly on estimating the index coefficients, the mean function, and their asymptotic properties. For accurate statistical inference it is equally important to estimate the error variance of these models. We examine two estimators of the error variance in a single‐index model and compare them with a few competing estimators with respect to their corresponding asymptotic properties. Using a simulation study, we evaluate the finite‐sample performance of our estimators against their competitors.     

非线性回归模型参数估计方法研究——以C-D生产函数为例

通过理论分析和蒙特卡罗模拟,对C-D生产函数模型参数的估计方法进行比较研究的结果表明：当误差项满足经典假设时,非线性最小二乘估计量具有与线性最小二乘估计类似的、近似BLUE的特性,且当误差项存在异方差时,用加权非线性最小二乘法也能大大改善估计量的性质。     

Shrinkage and Penalty Estimators of a Poisson Regression Model

In this paper we propose Stein‐type shrinkage estimators for the parameter vector of a Poisson regression model when it is suspected that some of the parameters may be restricted to a subspace. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Furthermore, we consider three different penalty estimators: the LASSO, adaptive LASSO, and SCAD estimators and compare their relative performance with that of the shrinkage estimators. Monte Carlo simulation studies reveal that the shrinkage strategy compares favorably to the use of penalty estimators, in terms of relative mean squared error, when the number of inactive predictors in the model is moderate to large. The shrinkage and penalty strategies are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Two Kinds of Restricted Estimators in Singular Linear Regression

In this article, the parameter estimators in singular linear model with linear equality restrictions are considered. The restricted root estimator and the generalized restricted root estimator are proposed and some properties of the estimators are also studied. Furthermore, we compare them with the restricted unified least squares estimator and show their sufficient conditions under which their superior over the restricted unified least squares estimator in terms of mean squares error, and discuss the choice of the unknown parameters of the generalized restricted root estimator.     

Innovational Outliers in INAR(1) Models

We consider integer-valued autoregressive models of order one contaminated with innovational outliers. Assuming that the time points of the outliers are known but their sizes are unknown, we prove that Conditional Least Squares (CLS) estimators of the offspring and innovation means are strongly consistent. In contrast, CLS estimators of the outliers' sizes are not strongly consistent. We also prove that the joint CLS estimator of the offspring and innovation means is asymptotically normal. Conditionally on the values of the process at time points preceding the outliers' occurrences, the joint CLS estimator of the sizes of the outliers is asymptotically normal.     

Nonlinear regression analysis for complex surveys 1

In a clustered finite population, it is assumed that a given function depending on an unknown parameter may be adopted to reveal the relationship among the variables of interest. The finite population parameter corresponding to this unknown parameter is defined as a solution of an estimating equation defined by a properly chosen population loss function. An estimation procedure that takes sample weights into account is considered. Use of this function in estimating the population mean per cluster is discussed. Large sample properties of estimators are investigated.     

Linear-representation Based Estimation of Stochastic Volatility Models

Abstract.  A new way of estimating stochastic volatility models is developed. The method is based on the existence of autoregressive moving average (ARMA) representations for powers of the log-squared observations. These representations allow to build a criterion obtained by weighting the sums of squared innovations corresponding to the different ARMA models. The estimator obtained by minimizing the criterion with respect to the parameters of interest is shown to be consistent and asymptotically normal. Monte-Carlo experiments illustrate the finite sample properties of the estimator. The method has potential applications to other non-linear time-series models.     

Estimating and Validating Nonlinear Regression Metamodels in Simulation

Frequently, the main objective of statistically designed simulation experiments is to estimate and validate regression metamodels, where the regressors are functions of the design variables and the dependent variable is the system response. In this article, a weighted least squares procedure for estimating the unknown parameters of a nonlinear regression metamodel is formulated and evaluated. Since the validity of a fitted regression model must be tested, a method for validating nonlinear regression simulation metamodels is presented. This method is a generalization of the cross-validation test proposed by Kleijnen (1983 Kleijnen , J. P. C. ( 1983 ). Cross-validation using the t statistic . European Journal of Operational Research 13 : 133 – 141 .[Crossref] , [Google Scholar]) in the context of linear regression metamodels. One drawback of the cross-validation strategy is the need to perform a large number of nonlinear regressions, if the number of experimental points is large. In this article, cross-validation is implemented using only one nonlinear regression. The proposed statistical analysis allows us to obtain Scheffé-type simultaneous confidence intervals for linear combinations of the metamodel's unknown parameters. Using the well-known M/M/1 example, a metamodel is built and validated with the aid of the proposed procedure.     

Monte carlo comparison of estimation methods for additive two-way tables

We considered the problem of estimating effects in the following linear model for data arranged in a two-way table: Response = Common effect + Row effect + Column effect + Residual. This work was occasioned by a project to analyse Federal Aviation Administration (FAA) data on daily temporal deviations from flight plans for commercial US flights, with rows and columns representing origin and destination airports, respectively. We conducted a large Monte Carlo study comparing the accuracy of three methods of estimation: classical least squares, median polish and least absolute deviations (LAD). The experiments included a wide spectrum of tables of different sizes and shapes, with different levels of non-linearity, noise variance, and percentages of empty cells and outliers. We based our comparison on the accuracy of the estimates and on computational speed. We identified factors that significantly affect accuracy and speed, and compared the methods based on their sensitivity to these factors. We concluded that there is no dominant method of estimation and identified conditions under which each method is most attractive.     

Small-sample behavior of weighted least squares in experimental design applications

In experimental design applications unbiased estimators s_i ² of the variances σ_i ² are possible. These estimators may be used in Weighted Least Squares (WLS) when estimating the parameters β. The resulting small-sample behavior is investigated in a Monte Carlo experiment. This experiment shows that an asymptotically valid covariance formula can be used if s_i ² is based on, say, at least 5 observations. The WLS estimator based on estimators s_i ² gives more accurate estimators of β, provided the σ_i ² differ by a factor, say, 10.     

A Simulation Study of Ridge Regression Estimators with Autocorrelated Errors

In this paper we run a large number of simulations to study the effects of collinearity and autocorrelated disturbances in the performance of several Ridge Regression estimators. The results suggest that with a fair amount of multicollinearity and of autocorrelation the Ridge Regression estimators which take the autocorrelation into account can perform better than the other methods. Also if the error term is only moderately autocorrelated; then the performance of the Ridge Regression estimators built upon ignoring the autocorrelation can outperform the other estimators.     

Bridge Estimation for Linear Regression Models with Mixing Properties

Penalized regression methods have for quite some time been a popular choice for addressing challenges in high dimensional data analysis. Despite their popularity, their application to time series data has been limited. This paper concerns bridge penalized methods in a linear regression time series model. We first prove consistency, sparsity and asymptotic normality of bridge estimators under a general mixing model. Next, as a special case of mixing errors, we consider bridge regression with autoregressive and moving average (ARMA) error models and develop a computational algorithm that can simultaneously select important predictors and the orders of ARMA models. Simulated and real data examples demonstrate the effective performance of the proposed algorithm and the improvement over ordinary bridge regression.     

Verallgemeinerte bayes-verfahren bei sequentiellen testproblemen

In this paper, problems of sequential decision theory are taken into consideration by extending the definition of the BAYES rule and treating BAYES rules. This generalisation is quite useful for practice. In many cases only BAYES rules can be calculated. The conditions under which such sequential decision procedures exist are demonstrated, as well as how to construct them on a scheme of backward induction resulting in the conclusion that the existence of BAYES rules needs essentially weaker assumptions than the existence of BAYES rules.Futhermore, methods are searched to simplify the construction of optimal stopping rules. Some illustrative examples are given.     

Combining Unbiased Ridge and Principal Component Regression Estimators

In the presence of multicollinearity problem, ordinary least squares (OLS) estimation is inadequate. To circumvent this problem, two well-known estimation procedures often suggested are the unbiased ridge regression (URR) estimator given by Crouse et al. (1995 Crouse , R. , Jin , C. , Hanumara , R. ( 1995 ). Unbiased ridge estimation with prior information and ridge trace . Commun. Statist. Theor. Meth. 24 : 2341 – 2354 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and the (r, k) class estimator given by Baye and Parker (1984 Baye , M. , Parker , D. ( 1984 ). Combining ridge and principal component regression: a money demand illustration . Commun. Statist. Theor. Meth. 13 : 197 – 205 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this article, we proposed a new class of estimators, namely modified (r, k) class ridge regression (MCRR) which includes the OLS, the URR, the (r, k) class, and the principal components regression (PCR) estimators. It is based on a criterion that combines the ideas underlying the URR and the PCR estimators. The standard properties of this new class estimator have been investigated and a numerical illustration is done. The conditions under which the MCRR estimator is better than the other two estimators have been investigated.     

基于LSTAR模型的非线性协整检验

大量的经济理论和实践都表明,宏观经济时间序列经常会出现非平稳和非线性特征,因而在统计分析时,需要进行非线性协整检验。基于逻辑平滑转换自回归(LSTAR)模型将传统的线性协整表述方法拓展为非线性形式,构造实用的检验程序及合适的统计量,利用软件R进行蒙特卡洛模拟给出非线性协整检验统计量的临界值,并通过实际数据分析购买力平价动态系统的非线性协整关系,说明方法的有效性。     

Isotonic Window Estimators of the Baseline Hazard Function in Cox's Regression Model Under Order Restriction

The regression model suggested by Cox (1972) has been widely used in survival analysis with censored observations. We propose isotonic window estimators for a monotone baseline hazard function in the Cox regression model. We prove that these estimators are asymptotically normal. The simulati on results presented in the article suggest that the proposed estimator performs better than several existing estimators in the literature     

Quantile Regression Estimator for GARCH Models

Abstract. In this article, we study the quantile regression estimator for GARCH models. We formulate the quantile regression problem by a reparametrization method and verify that the obtained quantile regression estimator is strongly consistent and asymptotically normal under certain regularity conditions. We also present our simulation results and a real data analysis for illustration.     

Asymptotic properties of a rank estimate in linear regression with symmetric non-identically distributed errors

In this article, a simple linear regression model with independent and symmetric but non-identically distributed errors is considered. Asymptotic properties of the rank regression estimate defined in Jaeckel [Estimating regression coefficients by minimizing the dispersion of the residuals, Ann. Math. Statist. 43 (1972), pp. 1449–1458] are studied. We show that the studied estimator is consistent and asymptotically normally distributed. The cases of bounded and unbounded score functions are examined separately. The regularity conditions of the article are exemplified for finite mixture distributions.