Simultaneous confidence bands for nonparametric regression with equal cluster size. (English) Zbl 07853640
Summary: Repeated/clustered data are becoming increasingly popular in many research studies. While there is a considerable literature in the estimation of the nonparametric component functions, the problem has been addressed by very few in statistical inference such as simultaneous confidence bands (SCBs). Based on the efficient kernel estimation incorporating within-cluster error correlation, we provide asymptotic Wald-type-based SCBs for the nonparametric function when the repeated measures have the equal cluster size. The performance of the SCBs is evaluated by simulation studies which support the asymptotic theory. The proposed method is also applied to a dataset on the Junior School Project of test scores.
{© 2022 John Wiley & Sons Ltd.}
{© 2022 John Wiley & Sons Ltd.}
MSC:
| 62-XX | Statistics |
Keywords:
kernel; nonparametric; repeated measure; simultaneous confidence bands; statistical inferenceSoftware:
ConfBandsReferences:
| [1] | Cai, L., Liu, R., Wang, S., & Yang, L. (2019). Simultaneous confidence bands for mean and variance functions based on deterministic design. Statistica Sinica, 29(1), 505-525. · Zbl 1412.62213 |
| [2] | Cao, G., Yang, L., & Todem, D. (2012). Simultaneous inference for the mean function based on dense functional data. Journal of Nonparametric Statistics, 24(2), 359-377. · Zbl 1241.62119 |
| [3] | Carroll, R. J., Maity, A., Mammen, E., & Yu, K. (2009). Nonparametric additive regression for repeatedly measured data. Biometrika, 96(2), 383-398. · Zbl 1163.62028 |
| [4] | Degras, D. (2017). Simultaneous confidence bands for the mean of functional data. WIREs Comput Stat, e1397, 9. · Zbl 07914924 |
| [5] | Gu, L., Wang, L., Härdle, W., & Yang, L. (2014). A simultaneous confidence corridor for varying coefficient regression with sparse functional data. Test, 23(4), 806-843. · Zbl 1312.62051 |
| [6] | Gu, L., Wang, S., & Yang, L. (2019). Simultaneous confidence bands for the distribution function of a finite population in stratified sampling. Annals of the Institute of Statistical Mathematics, 71(4), 983-1005. · Zbl 1426.62121 |
| [7] | Gu, L., & Yang, L. (2015). Oracally efficient estimation for single‐index link function with simultaneous confidence band. Electronic Journal of Statistics, 9(1), 1540-1561. · Zbl 1327.62254 |
| [8] | Härdle, W. (1989). Asymptotic maximal deviation of M‐smoothers. Journal of Multivariate Analysis, 29, 163-179. · Zbl 0667.62028 |
| [9] | Huang, J. Z., Zhang, L., & Zhou, L. (2007). Efficient estimation in marginal partially linear models for longitudinal/clustered data using splines. Scandinavian Journal of Statistics, 34(3), 451-477. · Zbl 1150.62020 |
| [10] | Krivobokova, T., Kneib, T., & Claeskens, G. (2010). Simultaneous confidence bands for penalized spline estimators. Journal of the American Statistical Association, 105(490), 852-863. · Zbl 1392.62094 |
| [11] | Li, J., & Yang, L. (2022). Statistical inference for functional time series. Statistica Sinica. https://doi.org/10.5705/ss.202021.0107 · Zbl 1540.62222 · doi:10.5705/ss.202021.0107 |
| [12] | Lin, X., & Carroll, R. (2000). Nonparametric function estimation for clustered data when the predictor is measured without/with error. Journal of the American Statistical Association, 95(450), 520-534. · Zbl 0995.62043 |
| [13] | Lin, X., Wang, N., Welsh, A., & Carroll, R. (2004). Equivalent kernels of smoothing splines in nonparametric regression for clustered/longitudinal data. Biometrika, 91(1), 177-193. · Zbl 1132.62321 |
| [14] | Liu, R., Yang, L., & Härdle, W. (2013). Oracally efficient two‐step estimation of generalized additive model. Journal of the American Statistical Association, 108, 619-631. · Zbl 1534.62054 |
| [15] | Ma, S. (2012). Two‐step spline estimating equations for generalized additive partially linear models with large cluster sizes. Annals of Statistics, 40(6), 2943-2972. · Zbl 1296.62093 |
| [16] | Ma, S., Song, Q., & Wang, L. (2013). Simultaneous variable selection and estimation in semiparametric modeling of longitudinal/clustered data. Bernoulli, 19(1), 252-274. · Zbl 1259.62021 |
| [17] | Mortimore, P., Sammons, P., Stoll, L., Lewis, D., & Ecob, R. (1989). A study of effective junior schools. International Journal of Educational Research, 13, 753-768. |
| [18] | Song, Q., & Yang, L. (2010). Oracally efficient spline smoothing of nonlinear additive autoregression model with simultaneous confidence band. Journal of Multivariate Analysis, 101(9), 2008-2025. · Zbl 1203.62161 |
| [19] | Wang, J., Cheng, F., & Yang, L. (2013). Smooth simultaneous confidence bands for cumulative distribution functions. Journal of Nonparametric Statistics, 25(2), 395-407. · Zbl 1297.62082 |
| [20] | Wang, J., Liu, R., Cheng, F., & Yang, L. (2014). Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band. Annals of Statistics, 42(2), 654-668. · Zbl 1308.62096 |
| [21] | Wang, J., Wang, S., & Yang, L. (2016). Simultaneous confidence bands for the distribution function of a finite population and its superpopulation. Test, 25(4), 692-709. · Zbl 1391.62077 |
| [22] | Wang, L., Xue, L., Qu, A., & Liang, H. (2014). Estimation and model selection in generalized additive partial linear models for correlated data with diverging number of covariates. Annals of Statistics, 42(2), 592-624. · Zbl 1309.62077 |
| [23] | Wang, N. (2003). Marginal nonparametric kernel regression accounting for within‐subject correlation. Biometrika, 90(1), 43-52. · Zbl 1034.62035 |
| [24] | Xia, Y. (1998). Bias‐corrected confidence bands in nonparametric regression. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60, 797-811. · Zbl 0909.62043 |
| [25] | Xue, L., Qu, A., & Zhou, J. (2010). Consistent model selection for marginal generalized additive model for correlated data. Journal of the American Statistical Association, 105(492), 1518-1530. · Zbl 1388.62223 |
| [26] | Zhang, Y., & Yang, L. (2018). A smooth simultaneous confidence band for correlation curve. Test, 27(2), 247-269. · Zbl 1404.62048 |
| [27] | Zheng, S., Liu, R., Yang, L., & Härdle, W. (2016). Statistical inference for generalized additive models: simultaneous confidence corridors and variable selection. Test, 25, 607-626. · Zbl 1422.62155 |
| [28] | Zheng, S., Yang, L., & Härdle, W. (2014). A smooth simultaneous confidence corridor for the mean of sparse functional data. Journal of the American Statistical Association, 109(506), 661-673. · Zbl 1367.62151 |
| [29] | Zhu, Z., Wing, F.K., & He, X. (2008). On the asymptotics of marginal regression splines with longitudinal data. Biometrika, 95(4), 907-917. · Zbl 1437.62679 |
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