臺灣博碩士論文加值系統

在隨機取樣的無母數迴歸分析中,本文研究核迴歸函數折線圖 (kernel regression function polygons) 及區域線性脊迴歸估計量 (local linear ridge regression estimators) 的理論性質,以下是其摘要說明。
在實務上,整條迴歸函數的估計曲線,是以折線圖形式來呈現的。該迴歸函數估計曲線,是利用取樣區域之若干等距離分割點上所得出的核估計量,以及線性插補法建構而成,本文稱之為核迴歸函數折線圖。在導出其漸近積分均方差 (asymptotic integrated mean square error) 後,本文據以計算使之最小化的最佳核函數。結果顯示,只有當分割點間距較帶寬值 (bandwidth) 具有更小的階數 (order) 時,艾氏核函數 (Epanechnikov kernel) 是此折線圖的最佳核函數。其次,當兩者的階數相同時,本文用數值方法計算出最佳核函數;隨著分割點間距的增加,折線圖的最小漸近積分均方差會逐漸上升,因而估計效果逐漸變差,這無疑是減少計算負荷下的代價。最後,當分割點間距具有較大的階數時,均勻核函數 (uniform kernel) 成為最佳核函數。
為改善區域線性估計量在有限樣本場合,其條件變異數沒有上界 (unbounded) 的潛在缺失, Seifert 與 Gasser (1996) 結合脊迴歸的觀念和區域線性平滑法,提出了區域線性脊迴歸估計量。本文首先研究這個估計量的局部性質。結果顯示,在有限樣本下,若區域線性脊迴歸估計量使用建立在有限區間的核函數,則其均方差之上界為有限的常數。在漸近狀態下,則必須適當地選取脊迴歸參數 (ridge parameters),才能使它具有區域線性估計量的諸多優秀性質,否則它將付出扭曲其漸近偏量行為的代價,本文得出此時脊迴歸參數所應滿足的若干條件。其次,模擬研究顯示,在合理的樣本規模下,使用本文建議之交叉有效法 (cross validation) 選取脊迴歸參數及帶寬值,可使區域線性脊迴歸估計量比區域線性估計量,達到明顯較低的樣本期望積分方差 (sample mean integrated square error)。

In the field of random design nonparametric regression, we examine two kernel estimators involving, respectively, piecewise linear interpolation of kernel regression function estimates and local ridge regression. Efforts dedicated to understanding their properties bring forth the following main messages.
The kernel estimate of a regression function inherits its smoothness properties from the kernel function chosen by the
investigator. Nevertheless, practical regression function estimates are often presented in interpolated form, using the exact kernel estimates only at some equally spaced grids of points. The asymptotic integrated mean square error (AIMSE) properties of such polygon type estimate, namely kernel regression function polygons (KRFP), are investigated. Call the "optimal kernel" the minimizer of the AIMSE. Epanechnikov kernel is not the optimal kernel unless for the case that the distance between every two consecutive grids is of smaller order in magnitude than the bandwidth used by the kernel regression function estimator. If the distance and bandwidth are of the same order in magnitude, we obtain the optimal kernel from the class of degree-two polynomials through numerical calculations. In this case, the best AIMSE performances deteriorate as the distance is increased to reduce the computational effort. When the distance is of larger order
in magnitude than the bandwidth, then uniform kernel serves as the optimal kernel for KRFP.
Local linear estimator (LLE) has many attractive asymptotic features. In finite sample situations, however, its conditional variance may become arbitrarily large. To cope with this difficulty, which can translate into the spurious rough appearance of the regression function estimate when design becomes sparse or clustered, Seifert and Gasser (1996)suggest "ridging" the LLE and propose the local linear ridge regression estimator (LLRRE). In this dissertation, local and numerical properties of the LLRRE are studied. It is shown that its finite sample mean square errors, both conditional and unconditional, are bounded above by finite constants. If the ridge regression parameters are not selected properly, then the resulting LLRRE suffers some drawbacks. For example, it is asymptotically biased and has boundary effects, and fails to inherit the nice asymptotic bias quality of the LLE. Letting the ridge parameters depend on sample size and converge to 0 as the sample size increases, we are able to ensure LLRRE the nice asymptotic features of the LLE under some mild conditions. Simulation studies demonstrate that the LLRRE using cross-validated bandwidth and ridge parameters could have smaller sample mean integrated square error than the LLE using cross-validated bandwidth, in reasonable sample sizes.

目 錄
第 1 章 導論 1
1.1 無母數迴歸分析簡介 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 1
1.2 區域線性估計﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 3
1.3 實務上的迴歸函數估計曲﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 9
1.4 有限樣本下的區域線性估計量 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 11
1.5 結果摘要 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 13
第 2 章 核迴歸函數折線圖 15
2.1 前言 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 15
2.2 核迴歸函數折線圖 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 18
2.3 漸近積分均方差 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 20
2.4 最佳核函數及估計效果﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 22
2.5 證明 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 28
第 3 章 區域線性脊迴歸估計量 47
3.1 區域線性脊迴歸估計量 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 47
3.2 理論性質 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 52
3.3 模擬研究 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 61
3.4 證明 ﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒﹒ 68
參考文獻 88

Bierens, H.J. (1987) Kernel estimators of regression functions. Cambridge University Press: Advances in Econometrics.
Chu, C.K. and Marron, J.S. (1991), Choosing a kernel regression estimator(with discussion), Statistical Science, 6, 404-436.
Cleveland, W.S. (1979), Robust locally weighted regression and smoothing scatter plots. Journal of the American Statistical Association, 74, 829-836.
Collomb, G. (1981), Estimation non-parametrique de la regression: revue bilbliographique. International Statistical Review, 49, 75-93.
Epanechnikov, V.A. (1969), Nonparametric estimates of a multivariate probability density. Theory of Probability and Its Applications, 14, 153-158.
Eubank, R.L. (1988), Spline Smoothing and Nonparametric Regression. Marcel Dekker Inc., New York.
Fan, J. (1992), Design-adaptive nonparametric regression. Journal of the American Statistical Association, 87, 998-1004.
Fan, J. (1993), Local linear regression smoothers and their minimax e.ciencies. Annals of Statistics, 21, 196-216.
Fan, J. and Gijbels I. (1992), Variable bandwidth and local linear regression smoothers. Annals of Statistics, 20, 2008-2036.
Fan, J. and Gijbels I. (1996), Local Polynomial Modelling and Its Applications—Theory and Methodologies. London: Chapman and Hall.
Gasser, T. and M¨uller, H.G. (1979), Kernel estimation of regression function. in Smoothing Techniques for Curve Estimation (T. Gasser and M. Rosenblatt, eds.), pp. 23-68, Heidelberg: Springer Verlag.
Gasser, T. and M¨uller, H.G. (1984), Estimating regression functions and their derivatives by the kernel method. Scandinavian Journal of Statistics, 11, 171-185.
Hall, P. and Marron, J.S. (1997), On the role of shrinkage parameter in local linear smoothing. Probability Theory and Related Fields, 108, 495-516.
Haerdle, W. (1990), Applied Nonparametric Regression. Cambridge University Press, New York.
Haerdle, W. (1991), Smoothing Techniques: With Implementation in S. Springer Series in Statistics, Springer Verlag, Berlin.
Haerdle, W., Hall, P., and Marron, J.S. (1988), How far are automatically chosen regression smoothing parameters from their optimum? (with discussion)Journal of the American Statistical Association, 83, 86-101.
Haerdle, W. and Marron, J.S. (1983), The nonexistence of moments of some kernel regression estimators. North Carolina Institute of Statistics, Mimeo. Series, No. 1537.
Haerdle, W. and Marron, J.S. (1985), Optimal bandwidth selection in nonparametric regression function estimation. Annals of Statistics 13, 1465-1481.
Jones, M.C. (1989), Discretized and interpolated kernel density estimates. Journal of the American Statistical Association, 84, 733-741.
Jones, M.C., Samiuddin, A., Al-Harbey. A.H. and Maatouk, T.A.H. (1998), The edge frequency polygon. Biometrika, 85, 235-239.
Marron, J.S. (1988), Automatic smoothing parameter selection: A survey. Empirical Economics, 13, 187-208.
Marron, J.S. andWand, M.P. (1992), Exact mean integrated squared error. Annals of Statistics, 20, 712-736.
Montgomery, D.C. and Peck, E.A. (1982), Introduction to Linear Regression Analysis. New York: Wiley.
Mueller, H.G. (1988), Nonparametric Regression Analysis of Longitudinal Data. Lecture Notes in Statistics, Vol. 46. Springer, Berlin.
Nadaraya, E.A. (1964), On estimating regression. Theory of Probability and Its Applications, 10, 186-190.
Rice, J.,(1984), Bandwidth choice for nonparametric regression. Annals of Statistics, 12, 1215-1230.
Rosenblatt, M. (1969), Conditional probability density and regression estimates. In: Multivariate Analysis II (Krishnaiah, ed.), 25-31. Academic Press, New York.
Scott, D.W. (1992), Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York.
Seifert, B. and Gasser, T. (1996), Finite-sample variance of local polynomials: Analysis and solutions. Journal of the American Statistical Association, 91, 267-275.
Simonoff, J.S. (1996), Smoothing Methods in Statistics. Springer-Verlag, New York.
Stone, C.J. (1980). Optimal rates of convergence for nonparametric estimators. Annals of Statistics, 8, 1348-1360.
Stuart, A. and Ord, J.K. (1987), Kendall’s Advanced Theory of Statistics, Vol. I. New York: Oxford University Press.
Wand, M.P. and Jones, M.C. (1995), Kernel Smoothing, New York: Chapman and Hall.
Watson, G.S. (1964), Smoothing regression analysis. Sankhya, Ser. A 26, 359-372.
Wu, J.S. and Chu, C.K. (1992), Double smoothing for kernel estimators in nonparametric regression. Journal of Nonparametric Statistics, 1, 375-386.