@article{012cc3a478b34b78858ec13fb3a92925,

title = "On discrete Epanechnikov kernel functions",

abstract = "Least-squares cross-validation is commonly used for selection of smoothing parameters in the discrete data setting; however, in many applied situations, it tends to select relatively small bandwidths. This tendency to undersmooth is due in part to the geometric weighting scheme that many discrete kernels possess. This problem may be avoided by using alternative kernel functions. Specifically, discrete versions (both unordered and ordered) of the popular Epanechnikov kernel do not have rapidly decaying weights. The analytic properties of these kernels are contrasted with commonly used discrete kernel functions and their relative performance is compared using both simulated and real data. The simulation and empirical results show that these kernel functions generally perform well and in some cases demonstrate substantial gains in terms of mean squared error.",

keywords = "Cross-validation, Discrete kernel, Panel data, Smoothing",

author = "Chu, {Chi Yang} and Henderson, {Daniel J.} and Parmeter, {Christopher F.}",

note = "Funding Information: We would like to thank the editor, Ana Colubi, an anonymous associate editor, two anonymous referees, Subha Chakraborti, Anna Gotlib and Jennifer Stoever. We would also like to thank conference participants at the 6th Asian Meeting of the Econometric Society, the 22nd International Panel Data Conference, the 3rd International Association for Applied Econometrics Conference, the 24th Symposium of the Society for Nonlinear Dynamics and Econometrics, the 25th Annual Meeting of the Midwest Econometrics Group, and seminar participants at Shandong University. Chu{\textquoteright}s research is partially supported by the National Natural Science Foundation of China (Project No. 71503147 ). Appendix A See Figs. A.1–A.5 . Appendix B Following Equation (A.3) in Li and Racine (2003, p. 278) , a power series expansion for our proposed kernel can be represented as L ( X i , x , h ) = 1 d i x = 0 − h 2 ( c − 1 ) − c h 2 k + 1 d i x = 1 − h 2 ( c − 1 ) − c h 2 k − 1 1 − h 2 ( c − 1 ) − c h 2 + 1 d i x = 2 − h 2 ( c − 1 ) − c h 2 k − 2 1 − h 2 ( c − 1 ) − c h 2 2 + 1 d i x = 3 − h 2 ( c − 1 ) − c h 2 k − 3 1 − h 2 ( c − 1 ) − c h 2 3 + ⋯ = the 1st term + the 2nd term + the 3rd term + 1 d i x = 3 − h 2 ( c − 1 ) − c h 2 k − 3 1 c h 2 − ( c − 1 ) 3 ( λ + 1 ) 2 − 1 3 + ⋯ = the 1st term + the 2nd term + the 3rd term + 1 d i x = 3 − h 2 ( c − 1 ) − c h 2 k − 3 1 c h 2 − ( c − 1 ) 3 ( λ + 2 ) 3 λ 3 + ⋯ ≈ the 1st term + the 2nd term + the 3rd term + O p ( λ 3 ) where h = λ + 1 with λ ∈ [ 0 , ∞ ) . Note that when λ = 0 , − h 2 ( c − 1 ) − c h 2 = 1 and 1 − h 2 ( c − 1 ) − c h 2 = 0 , which correspond to ( 1 − λ ) = 1 and λ = 0 for the unordered Li and Racine or Aitchison and Aitken kernels. Therefore, although c cannot be moved out of the square brackets and h appears in the denominators, these do not seem to affect any subsequent derivations and proof (e.g., Lemmas A.1–A.5). Note that although c can be rearranged as an extra multiplicative constant for the Aitchison and Aitken kernel, its value still can affect the power series expansion of the product kernel. Appendix C Here we provide four scenarios for the multinomial and beta-binomial distributions ( Tables C8 and C9 ) and these are primarily used in Section 3.1 . In Table C9 , Scenario (I) with c = 125, 250, and 500 is used in Section 3.1.2 . Additionally, we provide the bandwidths used to calculate the results in Tables 1–5 in Section 3.3 ( Tables C10–C14 ). Publisher Copyright: {\textcopyright} 2017 Elsevier B.V.",

year = "2017",

month = dec,

doi = "10.1016/j.csda.2017.07.003",

language = "English (US)",

volume = "116",

pages = "79--105",

journal = "Computational Statistics and Data Analysis",

issn = "0167-9473",

publisher = "Elsevier",

}