Consider the following time-varying coefficient model,

where for and is a zero-mean correlated stochastic process that cannot be explained by the covariates. As in Hoover et al. [9] we regard the repeated observations, for and, as a random sample from model (1), that is,

where is the observed response, is the observed covariates for the ith subject at time, and is a zero-mean stochastic process with covariance function . For this model, we assume that the measurements on the responses for different subjects are independent, but may be correlated at different time points within each subject,’s are also independent for different subjects. Model (1) or (2) is a useful extension to the usual linear model for longitudinal/panel data analysis in Lindsey [12] , Jones [13] , Diggle, et al. [1] , Hand and Crower [14] , among others.

There are several methods for estimating the coefficient, for example, the smoothing spline method in Brumback and Rice [15] , Hoover et al. [9] , and Chiang et al. [16] , the kernel smoother in Hoover et al. [9] , Wu et al. [17] , Wu and Chiang [18] , and Chiang et al. [16] , and other methods such as the two-step estimation in Fan and Zhang [10] and the global smoothing procedure using basis function approximations in Huang et al. [11] . These methods are all based on the least squares principle and suffer from the shortcoming of non-robustness. It is worthy of developing a robust estimation and testing method for the model (1). Generally speaking, one can develop different testing methods for different estimating approaches. To introduce our method, we begin with the local polynomial estimation (see Hoover et al. [9] ), that is, to find’s to minimize

where, and d denotes the order of the polynomial used in smoothing.

The above local least squares based estimation is not robust against outliers and heavy tailed errors. To fix this problem, we propose to estimate the coefficient function by minimizing

where is an outlier resistant function. The coefficient functions’s is estimated by’s, the solutions of the above optimization problem. The resulting estimators are so-called the local M-type estimator of. If has a derivative, then the solutions of (4) solve the following equations:

for and.

The above method is in the same spirit of the local M-estimation studied for cross-sectional data in Fan and Jiang [19] and Jiang and Mack [20] . It can be shown that the estimator is -consistent under certain conditions. The estimator involves a selection of the outlier resistant function. Much work in the literature has demonstrated that the Huber’s -class, i.e., , is satisfactory for robust estimation in the location model and nonparametric regression (Huber [21] ; Jiang and Mack [20] ), where a method for determining the parameter k was studied in Jiang and Mack [20] . However, our experience shows that a rule of thumb for the choice of k, such as where s is the robust standard deviation of the residuals (Serfling [22] ), is simple and satisfactory in the present situation. Other choices of the outlier resistant function are also possible, such as which leads to the least absolute deviation estimation (Jiang et al., [23] ).

The Newton algorithm can be used to find the solutions to the Equations (5). If the initial values of the parameters for iteration are good enough, for example, from the least squares estimation, then the iterative solutions can be found in a few steps, which is theoretically verified in Jiang and Mack [20] ) for nonparametrically modeling dependent data. In our simulations, the least squares based estimators in (2) will be employed as the initial values.

The performance of the estimator in (4) depends on the smoothing parameter h. There are several approaches to the selection of the bandwidth, such as the cross-validation (CV) and generalized cross-validation (GCV) criteria in Hastie and Tibshirani [24] . We here extend the CV method to the present situation for determining the bandwidth h. Specifically, denote by the solution of (4) or (5) based on all the observations without the measurements for ith subject. Then the bandwidth h can be estimated by the minimizer of

where.

The minimization of (6) is time-consuming for simulations, even though it is not for real data analysis. We here suggest a pre-determination of before simulations. Specifically, the procedure is detailed as follows:

1) Generate several samples (20 for instance) from (2), then minimize (6) to get the for each sample

and compute the average of the’s;

2) Fix the bandwidth in each simulation to find the estimator in (4).

In step 1), the computational burden can be further reduced if one uses an easy-evaluated bandwidth as initial value, such as the one from the GCV criterion for the local least squares based estimator. Since the estimated bandwidths for the local least squares estimation and the local M-estimation are highly correlated in general, the minimization of will be achieved in a few iterations.

In this section, we compare the performance of the local M-estimation with the local least squares estimation. For the outlier-resistant function, we opt for the Huber -functions in our numerical study:

The rule of thumb in Section 2.1 for the choice of k will be employed.

Example 1. Consider the following model,

where, , and are respectively from models in such a way that, , , and. For simplicity, we used the time points which are equally spaced for each i. The errors are iid for all i and j. We considered the following four distributions for the errors:

(A1);

(A2);

(A3);

(A4).

We simulated 200 samples of size and from the model (9). The averages of and the 2.5% and 97.5% sample quantiles of among simulations were calculated at each time points. In addition, we also computed the mean absolute deviation error (MADE) of the estimators at the time points (for),

The average of values of the above MADE over simulations was calculated and reported in Table 1 for the error models (A1) and (A3). The results show that the local LS estimator and the local M-estimators have similar accuracy if the error is normal, but the latter performs better than the former if the error deviates away from normality. We display in Figure 1 the estimators of the coefficient functions along with the widths of the envelopes formed by pointwise 2.5% and 97.5% quantiles of the estimators among simulations, where is easier to estimate than and the estimated envelopes’ widths for are not reported for saving space. It is evident that both the local LS estimator and the local M-estimator have little biases, and the latter is much better than the former in terms of the envelopes’ widths when the error deviates away from the normal distribution.

We did simulations with the error distributions in (A2) and (A4). The results are similar and omitted for saving space.

Example 2. In this example, we consider the model (9) with much more complex structures for the covariates, the errors, and the coefficient functions than those in Example 1. Different from Example 1, the two co-

variates are now correlated, and the error are correlated within subjects. Specifically, , , and are from the following two-dimensional AR model,

where and. We used the same’s as in Example 1. The errors are iid for all different subjects but may be correlated within subjects, which were generated from the following three distributions:

(B1);

(B2), where;

(B3), where.

We conducted 400 simulations. In each simulation, a samples of size and was drawn. We calculated the estimators along with the related MADEs and envelopes. Table 2 reports the MADEs for the estimated coefficient functions. It is evidenced that the local M-estimator is better than its counterpart under the heavy tailed error in (B3). Both estimators have similar MADEs under the other two errors in (B1) and (B2).

Since is much simpler than, we now consider only the the estimated curves for. Figure 2 displays the estimated curves for along with the 95% confidence intervals under the error (B2), and the widths of the envelopes for under the errors in (B2) and (B3). The average of the estimator in Figure 2(a) and the corresponding 95% confidence intervals in Figure 2(b) show that the local M-estimator well captures the structure of the true curve. It is seen from Figure 2(c) and Figure 2(d) that the local M-estimator is better than the local LS estimator when the error deviates away from the norm distribution.

In this section, we conduct simulations to show that our conditional bootstrap method gives a good estimate of the null distribution, and to compare the powers of different nonparametric tests.

Example 3. To compare the powers of different tests, we consider the testing problem (7) for the model (9). The null model is the constant-coefficient model with, , and are the same as in Example 1. The error is generated from one of the following distributions:

(C1);

(C2);

(C3).

We use a sequence of alternative models, , to calculate the powers of the tests.

We conducted 400 simulations. The sample size is the same as in Example 1. The bootstrap replicates number is. Figure 3 displays the histograms and the estimated probability densities of the testing statistic based on the common kernel method under the null model. It is seen that the null distributions of the testing statistics are very close, which shows that in finite samples the two tests hold close levels and hence have approx-

imately equal type I errors. Figure 4 shows the powers of the tests based on the local LS and local M-estimators. It demonstrates that the proposed test is more powerful than that based on local LS-estimator when the error deviates away from the normal distribution. Both tests are approximately the same powerful under the normal error.

We here illustrate how to use the proposed method in practice. Consider the body-weight of male Wistar rats dataset in Brunner et al. ([30] , Table A11). For this dataset, the objective of the experience is to assess the toxicity of a drug on the body-weight evolution of Wistar rats. A group of ten rats was given a placebo, while a second group of ten was given a high dose of the drug. For each rat in the study, its body-weight was observed once a week over a period of 22 weeks. Figure 5(a) and Figure 5(b) displays the data curves for the two groups.

To check if the body-weights of the two test groups differ in their evolution over time, Brunner et al. [2] compared the time curves of the mean body-weights for both groups, and evaluated the ANOVA-type statistics. Their results do not support the conjecture of different body-weight evolutions in the two groups.

We are interested in investigating the conjecture. Since Figure 5(a) and Figure 5(b) exhibit the time-varying feature of the body-weights of rats, it seems reasonable to model the dataset via the following time-varying coefficient model:

where equals 1 if in the treatment group and equals 0 if in the control group. Then reflects the evolution of average body-weight for the control group, and reflects that for the treatment group.

Since in the beginning the average body-weight of the rats in the placebo group is bigger than that in the

treatment group, we subtract the average body-weight in each group from the body-weight of each subject. This will make the estimators of the coefficient functions be approximately zeros at the beginning. Figure 5(c) and Figure 5(d) report the estimators of and. It seems that the is time-varying but the is not, which plausibly reflects that the drug did not affect the average body-weights of rats.

Now we consider the following two hypothesis testing problems:

1), where is an unknown parameter vector. This is used to test if the average body-weights of rats evolve over time.

2). This checks if the drug affects the average body-weights of rats in study.

We used bootstrap replicates for computing the null distributions of the testing statistics. For, the P-values are 0.0117 and 0.005 respectively for the local LS-estimation based test and the local M-estimation based test. It seems that the null hypothesis does not hold, which means that the average body-weight of rats for each group changes over time. For, the P-values are 0.9633 and 0.8833 for the tests respectively based on the local LS and local M-estimation methods. This evidences that for the dataset the drug has no effect on the average body-weights of rats. This is consistent to the result of Brunner et al. [30] .