Abstract
In this paper, we propose a local linear estimator for the regression model \(Y=m(X)+\varepsilon \) based on the reciprocal inverse Gaussian kernel when the design variable is supported on \((0,\infty )\). The conditional mean-squared error of the proposed estimator is derived, and its asymptotic properties are thoroughly investigated, including the asymptotic normality and the uniform almost sure convergence. The finite sample performance of the proposed estimator is evaluated via simulation studies and a real data application. A comparison study with other existing estimation methods is also made, and the pros and cons of the proposed estimator are discussed.
Similar content being viewed by others
References
Chatterjee S, Handcock M, Simonoff J (1995) A casebook for a first course in statistics and data analysis. Wiley, New York
Chaubey Y, Laïb N, Sen A (2010) Generalised kernel smoothing for non-negative stationary ergodic processes. J Nonparametr Stat 22:973–997
Chen SX (1999) Beta kernel estimators for density functions. Comput Stat Data Anal 31:131–145
Chen SX (2000a) Beta kernel smoothers for regression curves. Stat Sin 10:73–91
Chen SX (2000b) Probability density function estimation using Gamma kernels. Ann Inst Stat Math 52:471–480
Cowling A, Hall P (1996) On pseudodata methods for removing boundary effects in kernel density estimation. J R Stat Soc Ser B (Methodol) 58:551–563
Fan JQ, Gijbels I (1996) Local polynomial modeling and its applications. Chapman and Hall, London
Funke B, Kawka R (2015) Nonparametric density estimation for multivariate bounded data using two non-negative multiplicative bias correction methods. Comput Stat Data Anal 92:148–162
Gasser T, Müller HG (1979) Kernel estimation of regression functions. Smooth Tech Curve Estim 757:23–68
Hirukawa M, Sakudo M (2014) Nonnegative bias reduction methods for density estimstion using asymmetric kernels. Comput Stat Data Anal 75:112–123
Igarashi G, Kakizawa Y (2014) Reformulation of inverse Gaussian, reciprocal inverse Gaussian, and Birnbaum–Saunders kernel estimators. Stat Probab Lett 84:235–246
Jin X, Kawczak J (2003) Birnbaum–Saunders and lognormal kernel estimators for modelling durations in high frequency financial data. Ann Econ Financ 4(1):103–124
John R (1984) Boundary modification for kernel regression. Commun Stat-Theory Methods 13:893–900
Kazuhisa M (2005), “Inverse Gaussian distribution. The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, NY, 10016-4309
Koul HL, Song WX (2013) Large sample results for varying kernel regression estimates. J Nonparametr Stat 25(4):829–853
Marchant C, Bertin K, Leiva V, Saulo H (2013) Generalized Birnbaum–Saunders kernel density estimators and an analysis of financial data. Comput Stat Data Anal 63:1–15
Mnatsakanov R, Sarkisian K (2012) Varying kernel density estimation on R+. Stat Probab Lett 82(7):1337–1345
Scaillet O (2004) Density estimation using inverse and reciprocal inverse Gaussian kernels. Nonparametr Stat 16:217–226
Shi JH, Song WX (2016) Asymptotic results in gamma kernel regression. Commun Stat-Theory Methods 45(12):3489–3509
Wahba G (1990) Spline models for observational data. Society for Industrial and Applied Mathematics Philadelphia, Pennsylvania
Wand MP, Marron JS, Ruppert D (1991) Transformations in density estimation. JASA 86:343–353
Wand MP, Jones MC (1994) Kernel smoothing, vol 60. Chapman & Hall/CRC, New York
Acknowledgements
This work is partially supported by the Natural Science Foundation of Shanxi Province, China(2013011002-1).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, X., Xiao, J., Song, W. et al. Local linear regression with reciprocal inverse Gaussian kernel. Metrika 82, 733–758 (2019). https://doi.org/10.1007/s00184-019-00717-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-019-00717-6