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Estimation and hypothesis test for partial linear multiplicative models. (English) Zbl 06957706
Summary: Estimation and hypothesis tests for partial linear multiplicative models are considered in this paper. A profile least product relative error estimation method is proposed to estimate unknown parameters. We employ the smoothly clipped absolute deviation penalty to do variable selection. A Wald-type test statistic is proposed to test a hypothesis on parametric components. The asymptotic properties of the estimators and test statistics are established. We also suggest a score-type test statistic for checking the validity of partial linear multiplicative models. The quadratic form of the scaled test statistic has an asymptotic chi-squared distribution under the null hypothesis and follows a non-central chi-squared distribution under local alternatives, converging to the null hypothesis at a parametric convergence rate. We conduct simulation studies to demonstrate the performance of the proposed procedure and a real data is analyzed to illustrate its practical usage.
MSC:
62 Statistics
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[1] Carroll, R. J.; Fan, J.; Gijbels, I.; Wand, M. P., Generalized partially linear single-index models, J. Amer. Statist. Assoc., 92, 438, 477-489, (1997) · Zbl 0890.62053
[2] Chen, K.; Guo, S.; Lin, Y.; Ying, Z., Least absolute relative error estimation, J. Amer. Statist. Assoc., 105, 491, 1104-1112, (2010) · Zbl 1390.62117
[3] Chen, K.; Lin, Y.; Wang, Z.; Ying, Z., Least product relative error estimation, J. Multivariate Anal., 144, 91-98, (2016) · Zbl 1328.62146
[4] Cook, R. D.; Weisberg, S., Residuals and influence in regression, (1982), Chapman and Hall New York · Zbl 0564.62054
[5] Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R., Least angle regression, Ann. Statist., 32, 2, 407-499, (2004), With discussion, and a rejoinder by the authors · Zbl 1091.62054
[6] Fan, J.; Li, R., Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc., 96, 456, 1348-1360, (2001) · Zbl 1073.62547
[7] Härdle, W.; Liang, H.; Gao, J. T., Partially linear models, (2000), Springer Physica Heidelberg · Zbl 0968.62006
[8] Heckman, N. E., Spline smoothing in partly linear models, J. R. Stat. Soc. Ser. B, 48, 244-248, (1986) · Zbl 0623.62030
[9] Liang, H.; Härdle, W.; Carroll, R. J., Estimation in a semiparametric partially linear errors-in-variables model, Ann. Statist., 27, 5, 1519-1535, (1999) · Zbl 0977.62036
[10] Liang, H.; Li, R., Variable selection for partially linear models with measurement errors, J. Amer. Statist. Assoc., 104, 234-248, (2009) · Zbl 1388.62208
[11] Liang, H.; Liu, X.; Li, R.; Tsai, C. L., Estimation and testing for partially linear single-index models, Ann. Statist., 38, 3811-3836, (2010) · Zbl 1204.62068
[12] Liang, H.; Wang, H.; Tsai, C. L., Profiled forward regression for ultrahigh dimensional variable screening in semiparametric partially linear models, Statist. Sinica, 22, 2, 531-554, (2012) · Zbl 1238.62045
[13] Müller, P.; van de Geer, S., The partial linear model in high dimensions, Scand. J. Stat., 42, 2, 580-608, (2015) · Zbl 1364.62196
[14] Speckman, P. E., Kernel smoothing in partial linear models, J. R. Stat. Soc. Ser. B, 50, 413-436, (1988) · Zbl 0671.62045
[15] Stute, W.; Zhu, L., Nonparametric checks for single-index models, Ann. Statist., 33, 1048-1083, (2005) · Zbl 1080.62023
[16] Tibshirani, R., Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B. Methodol., 58, 1, 267-288, (1996) · Zbl 0850.62538
[17] Wang, Q.; Jing, B., Empirical likelihood for partial linear models, Ann. Inst. Statist. Math., 55, 3, 585-595, (2003) · Zbl 1047.62026
[18] Wei, C.; Wang, Q., Statistical inference on restricted partially linear additive errors-in-variables models, Test, 21, 4, 757-774, (2012) · Zbl 1284.62286
[19] Xie, H.; Huang, J., Scad-penalized regression in high-dimensional partially linear models, Ann. Statist., 37, 2, 673-696, (2009) · Zbl 1162.62037
[20] Yang, Y.; Li, G.; Lian, H., Nonconcave penalized estimation for partially linear models with longitudinal data, Statistics, 50, 1, 43-59, (2016) · Zbl 1342.62052
[21] Zhou, Y.; Liang, H., Statistical inference for semiparametric varying-coefficient partially linear models with error-prone linear covariates, Ann. Statist., 37, 427-458, (2009) · Zbl 1156.62036
[22] Zhu, L.; Cui, H., Testing the adequacy for a general linear errors-in-variables model, Statist. Sinica, 15, 4, 1049-1068, (2005) · Zbl 1086.62026
[23] Zhu, L.; Ng, K. W., Checking the adequacy of a partial linear model, Statist. Sinica, 13, 3, 763-781, (2003) · Zbl 1028.62032
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