Detection of the symmetry of model errors for partial linear single-index models. (English) Zbl 1489.62125
Summary: In this paper, we propose a \(k\)-th correlation coefficient estimator between the density function and distribution function of the model errors in single-index models and partial linear single-models. This \(k\)-th correlation coefficient estimator is used to test whether the density function of the true model error is symmetric or not. First, we propose a moment based estimator of \(k\)-th correlation coefficient and present its asymptotic results. Second, we consider statistical inference of the \(k\)-th correlation coefficient estimator by using the empirical likelihood method. The empirical likelihood statistic is shown to be asymptotically distributed a centered chi-squared distribution with degree of freedom one. Simulation studies are conducted to examine the performance of the proposed estimators and test statistics.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 62G05 | Nonparametric estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G10 | Nonparametric hypothesis testing |
| 62E20 | Asymptotic distribution theory in statistics |
References:
| [1] | Bindele, H. F.; Abebe, A.; Meyer, K. N., General rank-based estimation for regression single index models, Annals of the Institute of Statistical Mathematics, 70, 5, 1115-46 (2018) · Zbl 1407.62129 · doi:10.1007/s10463-017-0618-9 |
| [2] | Carroll, R. J.; Härdle, W.; Liang, H., Estimation in a semiparametric partially linear errors-in-variables model, The Annals of Statistics, 27, 5, 1519-35 (1999) · Zbl 0977.62036 · doi:10.1214/aos/1017939140 |
| [3] | Csörgő, S.; Heathcote, C. R., Testing for symmetry, Biometrika, 74, 1, 177-84 (1987) · Zbl 0606.62049 · doi:10.1093/biomet/74.1.177 |
| [4] | Cui, X.; Guo, W.; Lin, L.; Zhu, L., Covariate-adjusted nonlinear regression, The Annals of Statistics, 37, 4, 1839-70 (2009) · Zbl 1168.62035 · doi:10.1214/08-AOS627 |
| [5] | Cui, X.; Härdle, W. K.; Zhu, L., The EFM approach for single-index models, The Annals of Statistics, 39, 3, 1658-88 (2011) · Zbl 1221.62062 · doi:10.1214/10-AOS871 |
| [6] | Davis, C. E.; Quade, D., U-statistics for skewness or symmetry, Communications in Statistics - Theory and Methods, 7, 5, 413-8 (1978) · Zbl 0391.62033 · doi:10.1080/03610927808827634 |
| [7] | Eberl, A.; Klar, B., A note on a measure of asymmetry, Statistical Papers (2019) · Zbl 1458.60022 · doi:10.1007/s00362-019-01145-4 |
| [8] | Feng, S. Y.; Li, G.; Zhang, J., Efficient statistical inference for partially nonlinear errors-in-variables models, Acta Mathematica Sinica, English Series, 30, 9, 1606-20 (2014) · Zbl 1298.62111 · doi:10.1007/s10114-014-1358-x |
| [9] | Gai, Y.; Zhu, X.; Zhang, J., Testing symmetry of model errors for nonparametric regression models by using correlation coefficient, Communications in Statistics - Simulation and Computation (2019) · Zbl 1524.62176 · doi:10.1080/03610918.2019.1670844 |
| [10] | Gupta, M. K., An asymptotically nonparametric test of symmetry, The Annals of Mathematical Statistics, 38, 3, 849-66 (1967) · Zbl 0157.48102 · doi:10.1214/aoms/1177698879 |
| [11] | Hall, P., The bootstrap and edgeworth expansion (1992), New York: Springer-Verlag, New York · Zbl 0744.62026 |
| [12] | Härdle, W.; Liang, H.; Gao, J. T., Partially linear models (2000), Heidelberg: Springer Physica, Heidelberg · Zbl 0968.62006 |
| [13] | Hettmansperger, T. P.; McKean, J. W.; Sheather, S. J., IMS Lecture Notes Monogr. Ser, 42 of, Testing symmetry of the errors of a linear model, 99-112 (2003), Beachwood, OH: Institute of Mathematical Statistics, Beachwood, OH |
| [14] | Hill, D. L.; Rao, P. V., Tests of symmetry based on Cramér-von Mises statistics, Biometrika, 64, 3, 489-94 (1977) · Zbl 0376.62029 · doi:10.1093/biomet/64.3.489 |
| [15] | Horowitz, J. L., Semiparametric and Nonparametric Methods in Econometrics (2009), New York: Springer, New York · Zbl 1278.62005 |
| [16] | Hsieh, J.; Jeng, J.; Lin, Y.; Kuo, Y., Single index fuzzy neural networks using locally weighted polynomial regression, Fuzzy Sets and Systems, 368, 82-100 (2019) · doi:10.1016/j.fss.2019.02.010 |
| [17] | Ichimura, H., Semiparametric least squares (SLS) and weighted SLS estimation of single-index models, Journal of Econometrics, 58, 1-2, 71-120 (1993) · Zbl 0816.62079 · doi:10.1016/0304-4076(93)90114-K |
| [18] | Jia, Y.; Song, T.; Wu, S.; Zhang, Q.; Su, Y., Dynamic influence prediction of social network based on partial autoregression single index model, Discrete Dynamics in Nature and Society, 2019, 1-15 (2019) · Zbl 1453.91079 · doi:10.1155/2019/6237406 |
| [19] | Kraft, C. H.; van Eeden, C., Linearized rank estimates and signed-rank estimates for the general linear hypothesis, The Annals of Mathematical Statistics, 43, 1, 42-57 (1972) · Zbl 0238.62045 · doi:10.1214/aoms/1177692699 |
| [20] | Kulasekera, K.; Wang, J., A test of equality of regression curves using gâteaux scores, Australian & New Zealand Journal of Statistics, 43, 1, 89-99 (2001) · Zbl 0990.62039 · doi:10.1111/1467-842X.00157 |
| [21] | Lai, P.; Li, G.; Lian, H., Semiparametric estimation of fixed effects panel data single-index model, Statistics & Probability Letters, 83, 6, 1595-602 (2013) · Zbl 1278.62048 · doi:10.1016/j.spl.2013.03.005 |
| [22] | Lian, H., Empirical likelihood confidence intervals for nonparametric functional data analysis, Journal of Statistical Planning and Inference, 142, 7, 1669-77 (2012) · Zbl 1238.62057 · doi:10.1016/j.jspi.2012.02.008 |
| [23] | Liang, H.; Liu, X.; Li, R.; Tsai, C. L., Estimation and testing for partially linear single-index models, The Annals of Statistics, 38, 6, 3811-36 (2010) · Zbl 1204.62068 · doi:10.1214/10-AOS835 |
| [24] | Liang, H.; Qin, Y.; Zhang, X.; Ruppert, D., Empirical likelihood-based inferences for generalized partially linear models, Scandinavian Journal of Statistics, 36, 3, 433-43 (2009) · Zbl 1197.62092 · doi:10.1111/j.1467-9469.2008.00632.x |
| [25] | Lian, H.; Liang, H.; Carroll, R. J., Variance function partially linear single-index models, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77, 1, 171-94 (2015) · Zbl 1414.62301 · doi:10.1111/rssb.12066 |
| [26] | Li, G.; Lai, P.; Lian, H., Variable selection and estimation for partially linear single-index models with longitudinal data, Statistics and Computing, 25, 3, 579-93 (2015) · Zbl 1331.62336 · doi:10.1007/s11222-013-9447-8 |
| [27] | Li, G.; Peng, H.; Dong, K.; Tong, T., Simultaneous confidence bands and hypothesis testing for single-index models, Statist. Sinica, 24, 2, 937-55 (2014) · Zbl 1285.62044 |
| [28] | Li, G.; Zhang, J.; Feng, S., Modern measurement error models (2016), Beijing: Science Press, Beijing |
| [29] | Ma, S.; Zhang, J.; Sun, Z.; Liang, H., Integrated conditional moment test for partially linear single index models incorporating dimension-reduction, Electronic Journal of Statistics, 8, 1, 523-42 (2014) · Zbl 1348.62141 · doi:10.1214/14-EJS893 |
| [30] | Mack, Y. P.; Silverman, B. W., Weak and strong uniform consistency of kernel regression estimates, Zeitschrift f⏧r Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 61, 3, 405-15 (1982) · Zbl 0495.62046 · doi:10.1007/BF00539840 |
| [31] | Orlov, A. I., Testing the symmetry of a distribution, Theory of Probability & Its Applications, 17, 2, 357-61 (1973) · Zbl 0253.62026 · doi:10.1137/1117041 |
| [32] | Owen, A., Empirical likelihood for linear models, The Annals of Statistics, 19, 4, 1725-47 (1991) · Zbl 0799.62048 · doi:10.1214/aos/1176348368 |
| [33] | Owen, A. B., Empirical likelihood (2001), London: Chapman and Hall/CRC, London · Zbl 0989.62019 |
| [34] | Partlett, C.; Patil, P. N., Measuring asymmetry and testing symmetry, Annals of the Institute of Statistical Mathematics, 69, 2, 429-60 (2017) · Zbl 1362.62105 · doi:10.1007/s10463-015-0547-4 |
| [35] | Patil, P. N.; Patil, P. P.; Bagkavos, D., A measure of asymmetry, Statistical Papers, 53, 4, 971-85 (2012) · Zbl 1254.62015 · doi:10.1007/s00362-011-0401-6 |
| [36] | Peng, H.; Huang, T., Penalized least squares for single index models, Journal of Statistical Planning and Inference, 141, 4, 1362-79 (2011) · Zbl 1204.62070 · doi:10.1016/j.jspi.2010.10.003 |
| [37] | Randles, R. H.; Fligner, M. A. Policello; Wolfe, I. I., An asymptotically distribution-free test for symmetry versus asymmetry, Journal of the American Statistical Association, 75, 369, 168-72 (1980) · Zbl 0427.62024 · doi:10.1080/01621459.1980.10477448 |
| [38] | Rothman, E. D.; Woodroofe, M., A Cramér-von Mises type statistic for testing symmetry, The Annals of Mathematical Statistics, 43, 6, 2035-8 (1972) · Zbl 0276.62045 · doi:10.1214/aoms/1177690879 |
| [39] | Serfling, R. J., Approximation theorems of mathematical statistics (1980), New York: John Wiley & Sons Inc, New York · Zbl 0538.62002 |
| [40] | Srinivasan, R.; Godio, L. B., A Cramér-von Mises type statistic for testing symmetry, Biometrika, 61, 1, 196-8 (1974) · Zbl 0285.62020 · doi:10.1093/biomet/61.1.196 |
| [41] | van der Vaart, A. W.; Wellner, J. A., Weak convergence and empirical processes. with applications to statistics (1996), New York: Springer-Verlag, New York · Zbl 0862.60002 |
| [42] | van Eeden, C., An analogue, for signed rank statistics, of Jurečková’s asymptotic linearity theorem for rank statistics, The Annals of Mathematical Statistics, 43, 3, 791-802 (1972) · Zbl 0243.62027 · doi:10.1214/aoms/1177692545 |
| [43] | Wang, L.; Xue, L.; Qu, A.; Liang, H., Estimation and model selection in generalized additive partial linear models for correlated data with diverging number of covariates, The Annals of Statistics, 42, 2, 592-624 (2014) · Zbl 1309.62077 · doi:10.1214/13-AOS1194 |
| [44] | Wang, W.; Zhu, Z., Variable selection for the partial linear single-index model, Acta Mathematicae Applicatae Sinica, English Series, 33, 2, 373-88 (2017) · Zbl 1368.62058 · doi:10.1007/s10255-017-0666-1 |
| [45] | Wei, Z.; Zhu, L., Evaluation of value at risk: An empirical likelihood approach, Statistica Sinica, 20, 1, 455-68 (2010) · Zbl 1180.62153 |
| [46] | Wilcoxon, F., Probability tables for individual comparisons by ranking methods, Biometrics, 3, 3, 119-22 (1947) · doi:10.2307/3001946 |
| [47] | Xia, Y., Asymptotic distributions for two estimators of the single-index model, Econometric Theory, 22, 6, 1112-37 (2006) · Zbl 1170.62323 · doi:10.1017/S0266466606060531 |
| [48] | Xia, Y.; Härdle, W., Semi-parametric estimation of partially linear single-index models, Journal of Multivariate Analysis, 97, 5, 1162-84 (2006) · Zbl 1089.62050 · doi:10.1016/j.jmva.2005.11.005 |
| [49] | 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 · doi:10.1080/02331888.2015.1074232 |
| [50] | Yang, Y.; Li, G.; Tong, T., Corrected empirical likelihood for a class of generalized linear measurement error models, Science China Mathematics, 58, 7, 1523-36 (2015) · Zbl 1327.62233 · doi:10.1007/s11425-015-4976-6 |
| [51] | Yang, Y.; Tong, T.; Li, G., Simex estimation for single-index model with covariate measurement error, AStA Advances in Statistical Analysis, 103, 1, 137-61 (2019) · Zbl 1427.62022 · doi:10.1007/s10182-018-0327-6 |
| [52] | Yang, S.; Xue, L.; Li, G., Simultaneous confidence band for single-index random effects models with longitudinal data, Statistics & Probability Letters, 85, 6-14 (2014) · Zbl 1285.62026 · doi:10.1016/j.spl.2013.10.014 |
| [53] | Zhang, J., Estimation and variable selection for partial linear single-index distortion measurement errors models, Statistical Papers (2019) · Zbl 1477.62105 · doi:10.1007/s00362-019-01119-6 |
| [54] | Zhang, J.; Feng, Z., Partial linear single-index models with additive distortion measurement errors, Communications in Statistics - Theory and Methods, 46, 24, 12165-93 (2017) · Zbl 1384.62143 · doi:10.1080/03610926.2017.1291971 |
| [55] | Zhang, J.; Feng, Z.; Xu, P., Estimating the conditional single-index error distribution with a partial linear mean regression, Test, 24, 1, 61-83 (2015) · Zbl 1315.62033 · doi:10.1007/s11749-014-0395-1 |
| [56] | Zhang, J.; Gai, Y.; Cui, X.; Li, G., Measuring symmetry and asymmetry of multiplicative distortion measurement errors, Communications in Statistics - Simulation and Computation (2019) · doi:10.1214/18-BJPS432 |
| [57] | Zhang, J.; Gai, Y.; Lin, B.; Zhu, X., Nonlinear regression models with single© heteroscedasticity, Statistica Neerlandica, 73, 2, 292-316 (2019) · Zbl 07788779 · doi:10.1111/stan.12170 |
| [58] | Zhang, Q.; Li, X.; Wang, W.; Wang, L., Dynamic cluster analysis of dependent networks, Chinese Journal of Applied Probabilities and Statistics, 35, 4, 397 (2019) · Zbl 1449.62146 |
| [59] | Zhang, J.; Niu, C.; Li, G., Exploring the constant coefficient of a single-index variation, Brazilian Journal of Probability and Statistics, 33, 1, 57-86 (2019) · Zbl 1414.62139 · doi:10.1214/17-BJPS377 |
| [60] | Zhang, Q.; Wang, K.; Li, D.; Lin, L., Gmm and misspecification correction for misspecified models with diverging number of parameters, Acta Mathematicae Applicatae Sinica, English Series, 35, 4, 780-97 (2019) · Zbl 1452.62212 · doi:10.1007/s10255-019-0852-4 |
| [61] | Zhang, J.; Wang, T.; Zhu, L.; Liang, H., A dimension reduction based approach for estimation and variable selection in partially linear single-index models with high-dimensional covariates, Electronic Journal of Statistics, 6, 2235-73 (2012) · Zbl 1295.62046 · doi:10.1214/12-EJS744 |
| [62] | Zhang, J.; Yu, Y.; Zhu, L.; Liang, H., Partial linear single index models with distortion measurement errors, Annals of the Institute of Statistical Mathematics, 65, 2, 237-67 (2013) · Zbl 1440.62141 · doi:10.1007/s10463-012-0371-z |
| [63] | Zhang, J.; Zhang, J.; Zhu, X.; Lu, T., Testing symmetry based on empirical likelihood, Journal of Applied Statistics, 45, 13, 2429-54 (2018) · Zbl 1516.62053 · doi:10.1080/02664763.2017.1421917 |
| [64] | Zhang, J.; Zhou, N.; Sun, Z.; Li, G.; Wei, Z., Statistical inference on restricted partial linear regression models with partial distortion measurement errors, Statistica Neerlandica, 70, 4, 304-31 (2016) · Zbl 1528.62024 · doi:10.1111/stan.12089 |
| [65] | Zhu, X.; Guo, X.; Zhu, L., An adaptive-to-model test for partially parametric single-index models, Statistics and Computing, 27, 5, 1193-204 (2017) · Zbl 1505.62451 · doi:10.1007/s11222-016-9680-z |
| [66] | Zhu, L.; Lin, L.; Cui, X.; Li, G., Bias-corrected empirical likelihood in a multi-link semiparametric model, Journal of Multivariate Analysis, 101, 4, 850-68 (2010) · Zbl 1181.62039 · doi:10.1016/j.jmva.2009.08.009 |
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.
