Robust model selection for a semimartingale continuous time regression from discrete data

The paper considers the problem of estimating a periodic function in a continuous time regression model observed under a general semimartingale noise with an unknown distribution in the case when continuous observation cannot be provided and only discrete time measurements are available. Two specific types of noises are studied in detail: a non-Gaussian Ornstein–Uhlenbeck process and a time-varying linear combination of a Brownian motion and compound Poisson process. We develop new analytical tools to treat the adaptive estimation problems from discrete data. A lower bound for the frequency sampling, needed for the efficiency of the procedure constructed by discrete observations, has been found. Sharp non-asymptotic oracle inequalities for the robust quadratic risk have been derived. New convergence rates for the efficient procedures have been obtained. An example of the regression with a martingale noise exhibits that the minimax robust convergence rate may be both higher or lower as compared with the minimax rate for the “white noise” model. The results of Monte-Carlo simulations are given.     

Identifying nonlinear covariate effects in semimartingale regression models

Summary LetX _t be a semimartingale which is either continuous or of counting process type and which satisfies the stochastic differential equationdX _t=Y_t(t, Z_t) dt+dM_t, whereY andZ are predictable covariate processes,M is a martingale and is an unknown, nonrandom function. We study inference for by introducing an estimator for and deriving a functional central limit theorem for the estimator. The asymptotic distribution turns out to be given by a Gaussian random field that admits a representation as a stochastic integral with respect to a multiparameter Wiener process. This result is used to develop a test for independence ofX from the covariateZ, a test for time-homogeneity of , and a goodness-of-fit test for the proportional hazards model (t,z)=₁(t)a ₂(z) used in survival analysis.Research supported by the Army Research Office under Grant DAAL03-86-K-0094Research supported by the Air Force Office of Scientific Research under Contract F49620-85-C-0007     

LAD estimation for nonlinear regression models with randomly censored data

The least absolute deviations (LAD) estimation for nonlinear regression models with randomly censored data is studied and the asymptotic properties of LAD estimators such as consistency, boundedness in probability and asymptotic normality are established. Simulation results show that for the problems with censored data, LAD estimation performs much more robustly than the least squares estimation.     

PARAMETER ESTIMATION IN LINEAR REGRESSION MODELS FOR LONGITUDINAL CONTAMINATED DATA

The parameter estimation and the coefficient of contamination for the regression models with repeated measures are studied when its response variables are contaminated by another random variable sequence. Under the suitable conditions it is proved that the estimators which are established in the paper are strongly consistent estimators.     

Frontier estimation via kernel regression on high power-transformed data

We present a new method for estimating the frontier of a multidimensional sample. The estimator is based on a kernel regression on the power-transformed data. We assume that the exponent of the transformation goes to infinity while the bandwidth of the kernel goes to zero. We give conditions on these two parameters to obtain complete convergence and asymptotic normality. The good performance of the estimator is illustrated on some finite sample situations.     

Improved estimation of regression parameters in measurement error models

The problem of simultaneous estimation of the regression parameters in a multiple regression model with measurement errors is considered when it is suspected that the regression parameter vector may be the null-vector with some degree of uncertainty. In this regard, we propose two sets of four estimators, namely, (i) the unrestricted estimator, (ii) the preliminary test estimator, (iii) the Stein-type estimator and (iv) the postive-rule Stein-type estimator. In an asymptotic setup, properties of these estimators are studied based on asymptotic distributional bias, MSE matrices, and risks under a quadratic loss function. In addition to the asymptotic dominance of the Stein-type estimators, the paper contains discussion of dominating confidence sets based on the Stein-type estimation. Asymptotic analysis is considered based on a sequence of local alternatives to obtain the desired results.     

Difference based estimation for partially linear regression models with measurement errors

This paper is concerned with the estimating problem of the partially linear regression models where the linear covariates are measured with additive errors. A difference based estimation is proposed to estimate the parametric component. We show that the resulting estimator is asymptotically unbiased and achieves the semiparametric efficiency bound if the order of the difference tends to infinity. The asymptotic normality of the resulting estimator is established as well. Compared with the corrected profile least squares estimation, the proposed procedure avoids the bandwidth selection. In addition, the difference based estimation of the error variance is also considered. For the nonparametric component, the local polynomial technique is implemented. The finite sample properties of the developed methodology is investigated through simulation studies. An example of application is also illustrated.     

Efficient estimation of longitudinal data additive varying coefficient regression models

Acta Mathematicae Applicatae Sinica, English Series - We consider a longitudinal data additive varying coefficient regression model, in which the coefficients of some factors (covariates) are...     

分组数据下线性回归模型的参数估计

讨论了分组数据下线性回归模型参数的MLE的存在、唯一性.通过EM算法获得MLE的近似解.通过SEM算法获得MLE的渐近协方差阵.     

A variable selection method based on Tabu search for logistic regression models

A Tabu search method is proposed and analysed for selecting variables that are subsequently used in Logistic Regression Models. The aim is to find from among a set of m variables a smaller subset which enables the efficient classification of cases. Reducing dimensionality has some very well-known advantages that are summarized in literature. The specific problem consists in finding, for a small integer value of p, a subset of size p of the original set of variables that yields the greatest percentage of hits in Logistic Regression. The proposed Tabu search method performs a deep search in the solution space that alternates between a basic phase (that uses simple moves) and a diversification phase (to explore regions not previously visited). Testing shows that it obtains significantly better results than the Stepwise, Backward or Forward methods used by classic statistical packages. Some results of applying these methods are presented.     

A semiparametric Wald statistic for testing logistic regression models based on case-control data

We propose a semiparametric Wald statistic to test the validity of logistic regression models based on case-control data. The test statistic is constructed using a semiparametric ROC curve estimator and a nonparametric ROC curve estimator. The statistic has an asymptotic chisquared distribution and is an alternative to the Kolmogorov-Smirnov-type statistic proposed by Qin and Zhang in 1997, the chi-squared-type statistic proposed by Zhang in 1999 and the information matrix test statistic proposed by Zhang in 2001. The statistic is easy to compute in the sense that it requires none of the following methods: using a bootstrap method to find its critical values, partitioning the sample data or inverting a high-dimensional matrix. We present some results on simulation and on analysis of two real examples. Moreover, we discuss how to extend our statistic to a family of statistics and how to construct its Kolmogorov-Smirnov counterpart. This work was supported by the 11.5 Natural Scientific Plan (Grant No. 2006BAD09A04) and Nanjing University Start Fund (Grant No. 020822410110)     

Efficient estimation for semiparametric varying- coefficient partially linear regression models with current status data

This article considers a semiparametric varying-coefficient partially linear regression model with current status data. The semiparametric varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model that allows one to explore the possibly nonlinear effect of a certain covariate on the response variable. A Sieve maximum likelihood estimation method is proposed and the asymptotic properties of the proposed estimators are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. The convergence rate of the estimator for the unknown smooth function is obtained and the estimator for the unknown parameter is shown to be asymptotically efficient and normally distributed. Simulation studies are conducted to examine the small-sample properties of the proposed estimates and a real dataset is used to illustrate our approach.     

Improved estimation in multiple linear regression models with measurement error and general constraint

In this paper, we define two restricted estimators for the regression parameters in a multiple linear regression model with measurement errors when prior information for the parameters is available. We then construct two sets of improved estimators which include the preliminary test estimator, the Stein-type estimator and the positive rule Stein type estimator for both slope and intercept, and examine their statistical properties such as the asymptotic distributional quadratic biases and the asymptotic distributional quadratic risks. We remove the distribution assumption on the error term, which was generally imposed in the literature, but provide a more general investigation of comparison of the quadratic risks for these estimators. Simulation studies illustrate the finite-sample performance of the proposed estimators, which are then used to analyze a dataset from the Nurses Health Study.     

General rank-based estimation for regression single index models

This study considers rank estimation of the regression coefficients of the single index regression model. Conditions needed for the consistency and asymptotic normality of the proposed estimator are established. Monte Carlo simulation experiments demonstrate the robustness and efficiency of the proposed estimator compared to the semiparametric least squares estimator. A real-life example illustrates that the rank regression procedure effectively corrects model nonlinearity even in the presence of outliers in the response space.     

New tests for jumps in semimartingale models

In this paper we propose a test to determine whether jumps are present in a discretely sampled process or not. We use the concept of truncated power variation to construct our test statistics for (i) semimartingale models and (ii) semimartingale models with noise. The test statistics diverge to infinity if jumps are present and have a normal distribution otherwise. Our method is valid (under very weak assumptions) for all semimartingales with absolute continuous characteristics and rather general model for the noise process. We finally implement the test and present the simulation results. Our simulations suggest that for semimartingale models the new test is much more powerful than tests proposed by Barndorff-Nielsen and Shephard (J Fin Econ 4:1–30, 2006) and Aït-Sahalia and Jacod (Ann Stat 371:184–222, 2009).     

Stahel-Donoho kernel estimation for fixed design nonparametric regression models

This paper reports a robust kernel estimation for fixed design nonparametric regression models. A Stahel-Donoho kernel estimation is introduced, in which the weight functions depend on both the depths of data and the distances between the design points and the estimation points. Based on a local approximation, a computational technique is given to approximate to the incomputable depths of the errors. As a result the new estimator is computationally efficient. The proposed estimator attains a high breakdown point and has perfect asymptotic behaviors such as the asymptotic normality and convergence in the mean squared error. Unlike the depth-weighted estimator for parametric regression models, this depth-weighted nonparametric estimator has a simple variance structure and then we can compare its efficiency with the original one. Some simulations show that the new method can smooth the regression estimation and achieve some desirable balances between robustness and efficiency.     

Polynomial spline estimation for partial functional linear regression models

Because of its orthogonality, interpretability and best representation, functional principal component analysis approach has been extensively used to estimate the slope function in the functional linear model. However, as a very popular smooth technique in nonparametric/semiparametric regression, polynomial spline method has received little attention in the functional data case. In this paper, we propose the polynomial spline method to estimate a partial functional linear model. Some asymptotic results are established, including asymptotic normality for the parameter vector and the global rate of convergence for the slope function. Finally, we evaluate the performance of our estimation method by some simulation studies.     

Robust adaptive efficient estimation for semi-Markov nonparametric regression models

Statistical Inference for Stochastic Processes - We consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises. An adaptive model...     

Improved estimation in a non-Gaussian parametric regression

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The vector of unknown parameters is assumed to belong to a compact set. The noise is specified by the Ornstein–Uhlenbeck process driven by the mixture of a Brownian motion and a compound Poisson process. Improved estimates for the unknown regression parameters, based on a special modification of the James–Stein procedure with smaller quadratic risk than the usual least squares estimates, are proposed. The developed estimation scheme is applied for the improved parameter estimation in the discrete time regression with the autoregressive noise depending on unknown nuisance parameters.     

Finite mixture of regression models for censored data based on scale mixtures of normal distributions

Advances in Data Analysis and Classification - In statistical analysis, particularly in econometrics, the finite mixture of regression models based on the normality assumption is routinely used to...