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Empirical likelihood for nonparametric parts in semiparametric varying-coefficient partially linear models. (English) Zbl 1169.62028
Summary: Empirical-likelihood-based inference for the nonparametric parts in semiparametric varying-coefficient partially linear (SVCPL) models is investigated. An empirical log-likelihood approach to construct the confidence regions/intervals of the nonparametric parts is developed. An estimated empirical likelihood ratio is proved to be asymptotically standard $$\chi ^{2}$$-limited. A simulation study indicates that, compared with a normal approximation based approach and the bootstrap method, the proposed method described herein works better in terms of coverage probabilities and average areas/widths of confidence regions/bands. An application to a real data set is illustrated.

MSC:
 62G08 Nonparametric regression and quantile regression 62G15 Nonparametric tolerance and confidence regions 62G05 Nonparametric estimation 65C60 Computational problems in statistics (MSC2010) 62H12 Estimation in multivariate analysis 62E20 Asymptotic distribution theory in statistics
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References:
 [1] Cai, Z.; Fan, J.; Yao, Q., Functional-coefficient regression models for nonlinear time series, Journal of the American statistical association, 95, 941-956, (2000) · Zbl 0996.62078 [2] Chen, S.X.; Hall, P., Smoothed empirical likelihood confidence intervals for quantiles, The annals of statistics, 21, 1166-1181, (1993) · Zbl 0786.62053 [3] DiCiccio, T.J.; Hall, P.; Romano, J.P., Bartlett adjustment for empirical likelihood, The annals of statistics, 19, 1053-1061, (1991) · Zbl 0725.62042 [4] Fan, J.; Huang, T., Profile likelihood inferences on semiparametric varying-coefficient partially linear models, Bernoulli, 11, 1031-1057, (2005) · Zbl 1098.62077 [5] Fan, J.; Zhang, C.; Zhang, J., Generalized likelihood ratio statistics and wilks phenomenon, The annals of statistics, 29, 153-193, (2001) · Zbl 1029.62042 [6] Fan, J.; Zhang, J.T., Two-step estimation of functional linear models with applications to longitudinal data, Journal of the royal statistical society, series B, 62, 303-322, (2000) [7] Hall, P.; La Scala, B., Methodology and algorithms of empirical likelihood, International statistical review, 58, 109-127, (1990) · Zbl 0716.62003 [8] Härdie, W.; Liang, H.; Gao, J.T., Partially linear models, (2000), Springer-Verlag New York [9] Hastie, T.J.; Tibshirani, R., Varying-coefficient models, Journal of royal statistical association B, 55, 757-796, (1993) · Zbl 0796.62060 [10] Kolaczyk, E.D., Empirical likelihood for generalized linear models, Statistica sinica, 4, 199-218, (1994) · Zbl 0824.62062 [11] Liang, H.; Härdle, W.; Carroll, R.J., Estimation in a semiparametric partially linear errors-in-variables model, The annals of statistics, 27, 1519-1535, (1999) · Zbl 0977.62036 [12] Liang, H.; Wang, N., Partially linear single-index measurement error models, Statistica sinica, 15, 99-116, (2005) · Zbl 1061.62098 [13] Li, Q.; Huang, C.J.; Li, D.; Fu, T.T., Semiparametric smooth coefficient models, Journal of business and economic statistics, 20, 412-422, (2002) [14] Owen, A.B., Empirical likelihood ratio confidence intervals for a single function, Biometrika, 75, 237-249, (1988) · Zbl 0641.62032 [15] Owen, A.B., Empirical likelihood ratio confidence regions, The annals of statistics, 18, 90-120, (1990) · Zbl 0712.62040 [16] Owen, A.B., Empirical likelihood for linear models, The annals of statistics, 19, 1725-1747, (1991) · Zbl 0799.62048 [17] Owen, A.B., Empirical likelihood, (2001), Chapman & Hall/CRC Boca Raton, FL · Zbl 0989.62019 [18] Ruppert, D.; Wand, M.P., Multivariate weighted least squares regression, The annals of statistics, 22, 1346-1370, (1994) · Zbl 0821.62020 [19] Wahba, G., Partial spline models for semiparametric estimation of functions of several variables, (), pp. 319-329 [20] Wu, C.O.; Chiang, C.T.; Hoover, D.R., Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data, Journal of the American statistical association, 93, 1388-1402, (1998) · Zbl 1064.62523 [21] Xue, L.G.; Zhu, L.X., Empirical likelihood for a varying coefficient model with longitudinal data, Journal of the American statistical association, 102, 642-654, (2007) · Zbl 1172.62306 [22] You, J.H.; Zhou, Y., Empirical likelihood for semiparametric varying-coefficient partially linear regression models, Statistics & probability letters, 76, 412-422, (2006) · Zbl 1086.62057 [23] Zhang, W.; Lee, S.Y.; Song, X., Local polynomial Fitting in semivarying coefficient models, Journal of multivariate analysis, 82, 166-188, (2002) · Zbl 0995.62038
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