Abstract
We in this paper investigate smoothed score function based confidence regions for parameters in single-index models. Because a plug-in estimator of nonparametric link function causes the bias of smoothed score function to be non-negligible, the limit of the score function is asymptotically normal with a non-zero mean due to the slow convergence rate of nonparametric estimation. A bias-corrected smoothed score function is recommended for achieving centered normal limit without under-smoothing or high order kernel, and then the confidence region can be constructed by chi-square distribution. Simulation studies are carried out to assess the performance of bias-corrected local likelihood, and to compare with normal approximation approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carroll RJ, Fan J, Gijbels I, Wand MP (1997) Generalized partially linear single-index models. J Am Statist Ass 92: 477–489
Fan J, Gijbels I (1996) Local polynomial modeling and its applications. Chapman and Hall, London
Fan J, Gasser T, Gijbels I, Brockmann M, Engel J (1993) Local polynomial fitting: A stantard for nonparametric regression. Discussion paper 9315. Institut de Statistique, Universite Catholique de Louvain, Begium
Härdle W, Hall P, Ichimura H (1993) Optimal smoothing in single-index models. Ann Statist 21: 157–178
Manski CF (1975) Maximum score estimation of the stochastic utility model of choice. J Econom 3: 205–228
Manski CF (1985) Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. J Econom 27: 313–333
Murphy SA, van der Vaart AW (2000) On profile likelihood. J Am Statist Ass 95: 449–465
Powell JL, Stock JH, Stock TM (1989) Semiparametric estimation of index coefficients. Econemetrica 57: 1403–1430
Severini TA, Staniswalis JG (1994) Quasi-likelihood estimation in semiparametric models. J Am Statist Ass 89: 501–511
Severini TA, Wong WH (1992) Profile likelihood and conditionally parametric models. Ann Statist 20: 1768–1802
Stute W, Xue LG, Zhu LX (2007) Empirical likelihood inference in nonlinear errors-in-covariables models with validation data. J Am Statist Assoc 102: 332–346
Xue LG, Zhu LX (2007) Empirical likelihood for a varying coefficient model with longitudinal data. J Am Statist Assoc 102: 642–654
Zhu LX, Xue LG (2006) Empirical likelihood confidence region in a partially linear single-index model. J R Statist Soc B 68: 549–570
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is supported by NNSF project (10771123) of China, NBRP (973 Program 2007CB814901) of China, RFDP (20070422034) of China, NSF projects (Y2006A13 and Q2007A05) of Shandong Province of China and a grant from Research Grants Council of Hong Kong, Hong Kong, China.
Rights and permissions
About this article
Cite this article
Chen, Q., Lin, L. & Zhu, L. Bias-corrected smoothed score function for single-index models. Metrika 71, 45–58 (2010). https://doi.org/10.1007/s00184-008-0201-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-008-0201-8