Smoothing Spline Estimation for Skew-Symmetric Density Functions

偏斜化對稱分佈函數之曲線平滑估計

張升懋(Sheng-Mao Chang)；蘇南誠(Nan-Cheng Su)；嵇允嬋(Yunchan Chi)

Approximate cross validation ； Profile likelihood ； Skew-normal distribution ； Skew-symmetric distribution ； Smoothing spline ； 近似交叉驗證 ； 剖面概似函數 ； 偏斜化對稱分佈函數 ； 曲線平滑估計

中國統計學報

58卷1期（2020 / 03 / 01）

23 - 44

英文

The location-scale skew-symmetric distribution, consisting of two parts: a symmetric density function and a skew function, is attracting increasing attention, especially when observed data are obviously asymmetric. In this article, given the location and the scale parameter, we propose a smoothing spline estimation of the skew function, and then apply a profile likelihood approach to estimate the location and the scale. Moreover, an approximate cross-validation is derived to estimate relative Kullback-Leibler distances to alleviate the computation burden in choosing an optimal smoothing parameter for smoothing spline modeling. The proposed skew function estimator is twice differentiable and hence a Newton approach can be applied to find the maximum profile likelihood estimation. The performance of the proposed approach was examined by simulation and real data examples.

當資料呈現非對稱分佈時，考慮有關位置尺度族之偏斜化對稱分佈是一可行的方法。位置尺度族偏斜化對稱分佈函數由二部份組成，其一是一屬於尺度族的對稱分佈函數，其二則是一偏斜函數。本文採用最大化剖面概似函數方法估計位置及尺度參數，用無母數方法中之曲線平滑方法估計偏斜函數。此外，我們亦推導出相對Kullback-Leibler距離，用以簡化使用資料交叉驗證的計算量。對於曲線平滑方法，本文使用的核函數是二次可微，所以文獻上的大樣本性質及計算估計量的演算法得以套用。我們最後以電腦模擬及二個實證分析來展示本文所提出方法之可用性。

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