在無母數迴歸中對平均曲線的估計與檢定

Abstract

假設我們觀察一組重覆測量的黃料，其來自於有一平均曲線和某種誤差結構的母群體。在本文中，我們討論如何用無母數迴歸的方法來估計平均曲線和誤差結構。另外，我們也有興趣想知道二組重覆測量的資料是否來自於相同的一個母群體。首先，我們利用立方平滑樣條來估計平均函數，再用樣本變異矩陣的特徵向量及特徵值來估計誤差結構。而後，我們提出一個統計量，Tn，來檢定兩母體平均函數的異同。我們考慮資料為序列相關和來自於一個高斯過程這兩種特殊的例子。我們用蒙地卡羅方法找出Tn的實驗分佈，然後再找出拒絕域。在兩母體平均函數之差為常數時，我們將之與Hotelling T2 統計量相比較。我們也提出靴環法（bootstrap method）來分析實際的資料。最後，我們利用本文所提的方法來分析兩筆實際的資料。

Assume that we observe repeated measurements data from populations with unknown mean curves and covariance structures. In this article, we discuss how to estimate the unknown mean curves and covariance structures using nonparametric regression. Another interest is to test whether the mean curves of two populations are different. First, we use cubic smoothing splines to estimate mean curves. Then; we use the eigenvectors and eigenvalues of the sample covariance matrix to estimate the covariance structure. From these, we can get a statistic; Tn; to test the hypothesis of equal mean curves. For the cases that data are serially correlated and data come from a Gaussian process, we use Monte Carlo method to find the empirical distribution of Tn and then decide the critical region. When the difference between two mean curves is a constant, we compare the power of Tn with the traditional Hotelling T2 statistic at 0.95 significance level. We also propose a bootstrap method when analyzing the real data. Finally, we will deal with two real examples using the method developed in this paper.