×

Semi-parametric estimation of partially linear single-index models. (English) Zbl 1089.62050

Summary: One of the most difficult problems in applications of semi-parametric partially linear single-index models (PLSIM) is the choice of pilot estimators and complexity parameters which may result in radically different estimators. Pilot estimators are often assumed to be root-\(n\) consistent, although they are not given in a constructible way. Complexity parameters, such as a smoothing bandwidth, are constrained to a certain speed, which is rarely determinable in practical situations.
In this paper, efficient, constructible and practicable estimators of PLSIMs are designed with applications to time series. The proposed technique answers two questions of R. J. Carroll et al. [Generalized partially linear single-index models. J. Am. Stat. Assoc. 92, No. 438, 477–489 (1997; Zbl 0890.62053)]: no root-\(n\) pilot estimator for the single-index part of the model is needed and complexity parameters can be selected at the optimal smoothing rate. The asymptotic distribution is derived and the corresponding algorithm is easily implemented. Examples from real data sets (credit-scoring and environmental statistics) illustrate the technique and the proposed methodology of minimum average variance estimation (MAVE).

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G07 Density estimation
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 0890.62053
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arminger, G.; Euache, D.; Bonne, T., Analyzing credit risk data: a comparison of logistic discrimination, classification tree analysis, and feed forward subroles, Comput. Statist., 12, 293-310 (1997) · Zbl 0936.62115
[2] Bates, D. M.; Watts, D. G., Nonlinear Regression Analysis and Its Applications (1988), Wiley: Wiley New York · Zbl 0728.62062
[3] Bradley, R. C., Basic properties of strong mixing conditions, (Eberlein, E.; Taqqu, M. S., Dependence in Probability and Statistics: A Survey of Recent Results (1986), Birkhauser: Birkhauser Boston), 165-192
[4] Carroll, R. J.; Fan, J.; Gijbels, I.; Wand, M. P., Generalized partially linear single-index models, J. Amer. Statist. Assoc., 92, 477-489 (1997) · Zbl 0890.62053
[5] Fan, J.; Gijbels, I., Local Polynomial Modeling and Its Applications (1996), Chapman & Hall: Chapman & Hall London · Zbl 0873.62037
[6] Hall, P., On projection pursuit regression, Ann. Statist., 17, 573-588 (1989) · Zbl 0698.62041
[7] Härdle, W.; Hall, P.; Ichimura, H., Optimal smoothing in single-index models, Ann. Statist., 21, 157-178 (1993) · Zbl 0770.62049
[8] Härdle, W.; Janssen, P.; Serfling, R., Strong uniform consistency rates for estimators of conditional functionals, Ann. Statist., 16, 1428-1449 (1988) · Zbl 0672.62050
[9] Härdle, W.; Stoker, T. M., Investigating smooth multiple regression by method of average derivatives, J. Amer. Statist. Assoc., 84, 986-995 (1989) · Zbl 0703.62052
[10] W.E. Henley, D.J. Hand, A \(k\); W.E. Henley, D.J. Hand, A \(k\)
[11] M. Hristache, A. Juditsky, J. Polzehl, V. Spokoiny, Structure adaptive approach for dimension reduction, Ann. Statist. 29 (2001) 593-627.; M. Hristache, A. Juditsky, J. Polzehl, V. Spokoiny, Structure adaptive approach for dimension reduction, Ann. Statist. 29 (2001) 593-627. · Zbl 1043.62052
[12] Hristache, M.; Juditsky, A.; Spokoiny, V., Direct estimation of the single-index coefficients in single-index models, Ann. Statist., 29, 1537-1566 (2001)
[13] Ichimura, H.; Lee, L., Semiparametric least squares estimation of multiple index models: single equation estimation, (Barnett, W.; Powell, J.; Tauchen, G., Nonparametric and Semiparametric Methods in Econometrics and Statistics (1991), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0766.62065
[14] Li, K. C., Sliced inverse regression for dimension reduction (with discussion), Amer. Statist. Assoc., 86, 316-342 (1991)
[15] Linton, O., Second order approximation in the partially linear regression model, Econometrica, 63, 1079-1112 (1995) · Zbl 0836.62050
[16] Masry, E., Multivariate local polynomial regression for time series: uniform strong consistency and rates, J. Time Ser. Anal., 17, 571-599 (1996) · Zbl 0876.62075
[17] M. Müller, B. Rönz, Credit scoring using semiparametric methods, in: J. Franke, W. Härdle, G. Stahl (Eds.), Measuring Risk in Complex Stochastic Systems, Springer Lecture Notes in Statistics, vol. 147, Springer, Berlin, 2000, pp. 85-102.; M. Müller, B. Rönz, Credit scoring using semiparametric methods, in: J. Franke, W. Härdle, G. Stahl (Eds.), Measuring Risk in Complex Stochastic Systems, Springer Lecture Notes in Statistics, vol. 147, Springer, Berlin, 2000, pp. 85-102.
[18] Rio, E., The functional law of the iterated logarithm for stationary strongly mixing sequences, Ann. Probab., 23, 1188-1203 (1995) · Zbl 0833.60024
[19] Robinson, P. M., Root-\(N\)-Consistent semiparametric regression, Econometrica, 56, 931-954 (1988) · Zbl 0647.62100
[20] Ruppert, D.; Sheather, J.; Wand, P. M., An effective bandwidth selector for local least squares regression, J. Amer. Statist. Assoc., 90, 1257-1270 (1995) · Zbl 0868.62034
[21] Severini, T. A.; Staniswalis, I. G., Quasi-likelihood estimation in semiparametric models, J. Amer. Statist. Assoc., 89, 501-511 (1994) · Zbl 0798.62046
[22] Xia, Y.; Li, W. K., On single-index coefficient regression models, J. Amer. Statist. Assoc., 94, 1275-1285 (1999) · Zbl 1069.62548
[23] Xia, Y.; Tong, H.; Li, W. K., On extended partially linear single-index models, Biometrika, 86, 831-842 (1999) · Zbl 0942.62109
[24] Xia, Y.; Tong, H.; Li, W. K.; Zhu, L., An adaptive estimation of dimension reduction space (with discussions), J. Roy. Statist. Soc. B., 64, 1-28 (2002)
[25] Yu, Y.; Ruppert, D., Penalized spline estimation for partially linear single-index models, J. Amer. Statist. Assoc., 97, 1042-1054 (2002) · Zbl 1045.62035
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.