Zongwu Cai

Efficient Estimation and Inferences for Varying-Coefficient Models

Abstract This article deals with statistical inferences based on the varying-coefficient models proposed by Hastie and Tibshirani. Local polynomial regression techniques are used to estimate coefficient functions, and the asymptotic normality of the resulting estimators is established. The standard error formulas for estimated coefficients are derived and are empirically tested. A goodness-of-fit test technique, based on a nonparametric maximum likelihood ratio type of test, is also proposed to detect whether certain coefficient functions in a varying-coefficient model are constant or whether any covariates are statistically significant in the model. The null distribution of the test is estimated by a conditional bootstrap method. Our estimation techniques involve solving hundreds of local likelihood equations. To reduce the computational burden, a one-step Newton-Raphson estimator is proposed and implemented. The resulting one-step procedure is shown to save computational cost on an order of tens with no deterioration in performance, both asymptotically and empirically. Both simulated and real data examples are used to illustrate our proposed methodology.

Regression Quantiles For Time Series

In this paper we study nonparametric estimation of regression quantiles for time series data by inverting a weighted Nadaraya–Watson (WNW) estimator of conditional distribution function, which was first used by Hall, Wolff, and Yao (1999, Journal of the American Statistical Association 94, 154–163). First, under some regularity conditions, we establish the asymptotic normality and weak consistency of the WNW conditional distribution estimator for I±-mixing time series at both boundary and interior points, and we show that the WNW conditional distribution estimator not only preserves the bias, variance, and, more important, automatic good boundary behavior properties of local linear “double-kernel†estimators introduced by Yu and Jones (1998, Journal of the American Statistical Association 93, 228–237), but also has the additional advantage of always being a distribution itself. Second, it is shown that under some regularity conditions, the WNW conditional quantile estimator is weakly consistent and normally distributed and that it inherits all good properties from the WNW conditional distribution estimator. A small simulation study is carried out to illustrate the performance of the estimates, and a real example is also used to demonstrate the methodology.

Nonparametric Quantile Estimations For Dynamic Smooth Coefficient Models

In this article, quantile regression methods are suggested for a class of smooth coefficient time series models. We use both local polynomial and local constant fitting schemes to estimate the smooth coefficients in a quantile framework. We establish the asymptotic properties of both the local polynomial and local constant estimators for α-mixing time series. Also, a bandwidth selector based on the nonparametric version of the Akaike information criterion is suggested, together with a consistent estimate of the asymptotic covariance matrix. Furthermore, the asymptotic behaviors of the estimators at boundaries are examined. A comparison of the local polynomial quantile estimator with the local constant estimator is presented. A simulation study is carried out to illustrate the performance of estimates. An empirical application of the model to real data further demonstrates the potential of the proposed modeling procedures.

Local Linear Estimation for Time-Dependent Coefficients in Cox's Regression Models

This article develops a local partial likelihood technique to estimate the time-dependent coefficients in Coxs regression model. The basic idea is a simple extension of the local linear fitting technique used in the scatterplot smoothing. The coefficients are estimated locally based on the partial likelihood in a window around each time point. Multiple time-dependent covariates are incorporated in the local partial likelihood procedure. The procedure is useful as a diagnostic tool and can be used in uncovering time-dependencies or departure from the proportional hazards model. The programming involved in the local partial likelihood estimation is relatively simple and it can be modified with few efforts from the existing programs for the proportional hazards model. The asymptotic properties of the resulting estimator are established and compared with those from the local constant fitting. A consistent estimator of the asymptotic variance is also proposed. The approach is illustrated by a real data set from the study of gastric cancer patients and a simulation study is also presented.

Nonparametric Estimation Of Varying Coefficient Dynamic Panel Data Models

We suggest using a class of semiparametric dynamic panel data models to capture individual variations in panel data. The model assumes linearity in some continuous/discrete variables that can be exogenous/endogenous and allows for nonlinearity in other weakly exogenous variables. We propose a nonparametric generalized method of moments (NPGMM) procedure to estimate the functional coefficients, and we establish the consistency and asymptotic normality of the resulting estimators.

Weighted Nadaraya-Watson regression estimation

In this article, we study nonparametric estimation of regression function by using the weighted Nadaraya-Watson approach. We establish the asymptotic normality and weak consistency of the resulting estimator for [alpha]-mixing time series at both boundary and interior points, and we show that the weighted Nadaraya-Watson estimator not only preserves the bias, variance, and more importantly, automatic good boundary behavior properties of local linear estimator, but also makes computation fast. Furthermore, the asymptotic minimax efficiency is discussed. Finally, comparisons between weighted Nadaraya-Watson approach and local linear fitting are given.

Uniform strong estimation under α-mixing, with rates☆

Let s{;Xns};, n [greater-or-equal, slanted] 1, be a stationary [alpha]-mixing sequence of real-valued r.v.s with distribution function (d.f.) F, probability density function (p.d.f.) f and mixing coefficient [alpha](n). The d.f. F is estimated by the empirical d.f. Fn, based on the segment X1,..., Xn. By means of a mixingale argument, it is shown that Fn(x) converges almost surely to F(x) uniformly in x[set membership, variant]. An alternative approach, utilizing a Kiefer process approximation, establishes the law of the iterated logarithm for sups{;vb;Fn(x)-F(xvb;; x[set membership, variant]. The d.f. F is also estimated by a smooth estimate n, which is shown to converge almost surely (a.s.) to F, and the rate of convergence of sups{;vb;n(x) - F(x)vb;;; x[set membership, variant]s}; is of the order of O((log log n/n)). The p.d.f. f is estimated by the usual kernel estimate fn, which is shown to converge a.s. to f uniformly in x[set membership, variant], and the rate of this convergence is of the order of O((log log n/nh2n)), where hn is the bandwidth used in fn. As an application, the hazard rate r is estimated either by rn or n, depending on whether Fn or n is employed, and it is shown that rn(x) and n(x) converge a.s. to r(x), uniformly over certain compact subsets of , and the rate of convergence is again of the order of O((log log n/nh2n)). Finally, the rth order derivative of f, f(r), is estimated by f(r)n, and is shown that f(r)n(x) converges a.s. to f(r)(x) uniformly in x[set membership, variant].The rate of this convergence is of the order of O((log log n/nh2(r+1)n)).

Application of a local linear autoregressive model to BOD time series

In this paper, we analyze the biochemical oxygen demand data collected over two years from McDowell Creek, Charlotte, North Carolina, U.S.A., by fitting an autoregressive model with time-dependent coefficients. The local linear smoothing technique is developed and implemented to estimate the coefficient functions of the autoregressive model. A nonparametric version of the Akaike information criterion is developed to determine the order of the model and to select the optimal bandwidth. We also propose a hypothesis testing technique, based on the residual sum of squares and F-test, to detect whether certain coefficients in the model are really varying or whether any variables are significant. The approximate null distributions of the test are provided. The proposed model has some advantages, such as it is determined completely by data, it is easily implemented and it provides a better prediction. Copyright

Asymptotic properties of Kaplan-Meier estimator for censored dependent data

In some long term studies, a series of dependent and possibly censored failure times may be observed. Suppose that the failure times have a common marginal distribution function, and inferences about it are of interest to us. The main result of this paper is that, under certain regularity conditions, the Kaplan-Meier estimator can be expressed as the mean of random variables, with a remainder of some order. In addition, the asymptotic normality of the Kaplan-Meier estimator is derived.

Berry-esseen bounds for smooth estimator of a distribution function under association

Let {X n ; n≥1} be real-valued random variables forming a stationary sequence, and satisfying the basic requirement of being positively or negatively associated. Let F be the marginal distribution function of the X i s, which is estimated by a smooth kernel-type estimate , by means of the segment X 1,…,X n . The main result of this paper is that of providing, under certain regularity conditions, Berry-Esseen bounds for the smooth estimator , thereby providing asymptotic normality of with rates.