×

A simple approach to construct confidence bands for a regression function with incomplete data. (English) Zbl 1437.62244

Summary: A long-standing problem in the construction of asymptotically correct confidence bands for a regression function \(m(x)=E[Y|X=x]\), where \(Y\) is the response variable influenced by the covariate \(X\), involves the situation where \(Y\) values may be missing at random, and where the selection probability, the density function \(f(x)\) of \(X\), and the conditional variance of \(Y\) given \(X\) are all completely unknown. This can be particularly more complicated in nonparametric situations. In this paper, we propose a new kernel-type regression estimator and study the limiting distribution of the properly normalized versions of the maximal deviation of the proposed estimator from the true regression curve. The resulting limiting distribution will be used to construct uniform confidence bands for the underlying regression curve with asymptotically correct coverages. The focus of the current paper is on the case where \(X\in \mathbb{R}\). We also perform numerical studies to assess the finite-sample performance of the proposed method. In this paper, both mechanics and the theoretical validity of our methods are discussed.

MSC:

62J02 General nonlinear regression
62G15 Nonparametric tolerance and confidence regions
62D10 Missing data

Software:

np
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Burke, Murray D., A Gaussian Bootstrap Approach to Estimation and Tests, Asymptotic Methods in Probability and Statistics, 697-706 (1998) · Zbl 0922.62025
[2] Burke, M., Multivariate tests-of-fit and uniform confidence bands using a weighted bootstrap, Stat. Probab. Lett., 46, 13-20 (2000) · Zbl 0941.62065
[3] Cai, T.; Low, M.; Zongming, M., Adaptive confidence bands for nonparametric regression functions, J. Am. Stat. Assoc., 109, 1054-1070 (2014) · Zbl 1368.62093
[4] Claeskens, G.; Van Keilegom, I., Bootstrap confidence bands for regression curves and their derivatives, Ann. Stat., 31, 1852-1884 (2003) · Zbl 1042.62044
[5] Devroye, L.; Györfi, L.; Lugosi, G., A Probabilistic Theory of Pattern Recognition (1996), New York: Springer, New York · Zbl 0853.68150
[6] Eubank, Rl; Speckman, Pl, Confidence bands in nonparametric regression, J. Am. Stat. Assoc., 88, 424, 1287-1301 (1993) · Zbl 0792.62030
[7] Gu, L.; Yang, L., Oracally efficient estimation for single-index link function with simultaneous confidence band, Electron. J. Stat., 9, 1540-1561 (2015) · Zbl 1327.62254
[8] Györfi, L.; Kohler, M.; Krzyżak, A.; Walk, H., A Distribution-Free Theory of Nonparametric Regression (2002), New York: Springer, New York · Zbl 1021.62024
[9] Härdle, W., Asymptotic maximal deviation of M-smoothers, J. Multivar. Anal., 29, 163-179 (1989) · Zbl 0667.62028
[10] Härdle, W., Applied Nonparametric Regression (1990), Cambridge: Cambridge University Press, Cambridge · Zbl 0714.62030
[11] Härdle, W.; Song, S., Confidence bands in quantile regression, Econom. Theory, 26, 4, 1-22 (2010) · Zbl 1294.62145
[12] Hayfield, T.; Racine, J., Nonparametric econometrics: the np package, J. Stat. Softw., 27, 5, 1-32 (2008)
[13] Hollander, M.; Mckeague, Iw; Yang, J., Likelihood ratio-based confidence bands for survival functions, J. Am. Stat. Assoc., 92, 215-227 (1997) · Zbl 1090.62560
[14] Horváth, L., Approximations for hybrids of empirical and partial sums processes, J. Stat. Plan. Inference, 88, 1-18 (2000) · Zbl 0976.60044
[15] Horváth, L.; Kokoszka, P.; Steinebach, J., Approximations for weighted bootstrap processes with an application, Stat. Probab. Lett., 48, 59-70 (2000) · Zbl 0982.60019
[16] Horvitz, Dg; Thompson, Dj, A generalization of sampling without replacement from a finite universe, J. Am. Stat. Assoc., 47, 663-685 (1952) · Zbl 0047.38301
[17] Janssen, A., Resampling student’s t-type statistics, Ann. Inst. Stat. Math., 57, 507-529 (2005) · Zbl 1095.62050
[18] Janssen, A.; Pauls, T., How do bootstrap and permutation tests work?, Ann. Stat., 31, 768-806 (2003) · Zbl 1028.62027
[19] Johnston, Gj, Probabilities of maximal deviations for nonparametric regression function estimates, J. Multivar. Anal., 12, 402-414 (1982) · Zbl 0497.62038
[20] Kojadinovic, I.; Yan, J., Goodness-of-fit testing based on a weighted bootstrap: a fast large-sample alternative to the parametric bootstrap, Can. J. Stat., 40, 480-500 (2012) · Zbl 1349.62224
[21] Kojadinovic, I.; Yan, J.; Holmes, M., Fast large-sample goodness-of-fit for copulas, Stat. Sinica, 21, 841-871 (2011) · Zbl 1214.62049
[22] Konakov, Vd; Piterbarg, Vi, On the convergence rate of maximal deviation distribution, J. Multivar. Anal., 15, 279-294 (1984) · Zbl 0554.62034
[23] Liero, H., On the maximal deviation of the kernel regression function estimate, Ser. Stat., 13, 171-182 (1982) · Zbl 0494.62044
[24] Lei, Q.; Qin, Y., Confidence intervals for nonparametric regression functions with missing data: multiple design case, J. Syst. Sci. Complex., 24, 1204-1217 (2011) · Zbl 1383.62117
[25] Little, Rja; Rubin, Db, Statistical Analysis with Missing Data (2002), New York: Wiley, New York · Zbl 1011.62004
[26] Li, G.; Van Keilegom, I., Likelihood ratio confidence bands in nonparametric regression with censored data, Scand. J. Stat., 2, 547-562 (2002) · Zbl 1035.62030
[27] Mack, Yp; Silverman, Z., Weak and strong uniform consistency of kernel regression estimates, Z. Wahrsch. Verw. Gebiete, 61, 405-415 (1982) · Zbl 0495.62046
[28] Mason, Dm; Newton, Ma, A rank statistics approach to the consistency of a general bootstrap, Ann. Stat., 20, 1611-1624 (1992) · Zbl 0777.62045
[29] Massé, P.; Meiniel, W., Adaptive confidence bands in the nonparametric fixed design regression model, J. Nonparameter Stat., 26, 451-469 (2014) · Zbl 1329.62170
[30] Mojirsheibani, M.; Pouliot, W., Weighted bootstrapped kernel density estimators in two sample problems, J. Nonparameter Stat., 29, 61-84 (2017) · Zbl 1364.62085
[31] Mondal, S.; Subramanian, S., Simultaneous confidence bands for Cox regression from semiparametric random censorship, Lifetime Data Anal., 22, 122-144 (2016) · Zbl 1356.62188
[32] Muminov, Ms, On the limit distribution of the maximum deviation of the empirical distribution density and the regression function. I, Theory Probab. Appl., 55, 509-517 (2011) · Zbl 1397.62163
[33] Muminov, Ms, On the limit distribution of the maximum deviation of the empirical distribution density and the regression function II, Theory Probab. Appl., 56, 155-166 (2012) · Zbl 1397.62164
[34] Nadaraya, Ea, Remarks on nonparametric estimates for density functions and regression curves, Theory Probab. Appl., 15, 134-137 (1970) · Zbl 0228.62031
[35] Neumann, Mh; Polzehl, J., Simultaneous bootstrap confidence bands in nonparametric regression, J. Nonparameter Stat., 9, 307-333 (1998) · Zbl 0913.62041
[36] Praestgaard, J.; Wellner, Ja, Exchangeably weighted bootstraps of the general empirical process, Ann. Probab., 21, 2053-2086 (1993) · Zbl 0792.62038
[37] Proksch, K., On confidence bands for multivariate nonparametric regression, Ann. Inst. Stat. Math., 68, 209-236 (2016) · Zbl 1440.62133
[38] Qin, Y.; Qiu, T.; Lei, Q., Confidence intervals for nonparametric regression functions with missing data, Commun. Stat. Theory Methods, 43, 4123-4142 (2014) · Zbl 1305.62174
[39] Racine, J.; Li, Q., Cross-validated local linear nonparametric regression, Stat. Sinica, 14, 485-512 (2004) · Zbl 1045.62033
[40] Song, S.; Ritov, Y.; Härdle, W., Bootstrap confidence bands and partial linear quantile regression, J. Multivar. Anal., 107, 244-262 (2012) · Zbl 1236.62035
[41] Wandl, H.: On kernel estimation of regression functions. Wissenschaftliche Sitzungen zur Stochastik, WSS-03, Berlin. (1980)
[42] Wang, Q.; Shen, J., Estimation and confidence bands of a conditional survival function with censoring indicators missing at random, J. Multivar. Anal., 99, 928-948 (2008) · Zbl 1136.62379
[43] Wang, Q.; Qin, Y., Empirical likelihood confidence bands for distribution functions with missing responses, J. Stat. Plann. Inference, 140, 2778-2789 (2010) · Zbl 1188.62148
[44] Watson, Gs, Smooth regression analysis, Sankhya Ser. A, 26, 359-372 (1964) · Zbl 0137.13002
[45] Xia, Y., Bias-corrected confidence bands in nonparametric regression, J. R. Stat. Soc. Ser. B. Stat. Methodol., 60, 797-811 (1998) · Zbl 0909.62043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.