Estimating smooth monotone functions
Estimating smooth monotone functions
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DOI：
10.1111/1467-9868.00130
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发表时间：
1998-01-01
影响因子：
5.8
通讯作者：
Ramsay, JO
中科院分区：
文献类型：
--
作者：
Ramsay, JO
许多情况下要求光滑的严格单调函数f具有任意的灵活性。由微分方程D（2）f = wDf定义的函数族，其中w是无约束系数函数，包括严格单调的二次可微函数。该方程的解为f = C-0 + C-1 D-1{exp（D（-1）w）}，其中C-0和C-1是任意常数，D-1是部分积分算子。建议扩大W的基础，允许明确的F的表达中的整合。在拟合数据时，通过惩罚w（2）的积分来正则化f也很有用，因为这是f的相对曲率的度量。讨论了单调非参数回归、非线性回归中因变量的变换和密度估计的应用。
Many situations call for a smooth strictly monotone function f of arbitrary flexibility. The family of functions defined by the differential equation D(2)f = w Df, where w is an unconstrained coefficient function, comprises the strictly monotone twice differentiable functions. The solution to this equation is f = C-0 + C-1 D-1{exp(D(-1)w)}, where C-0 and C-1 are arbitrary constants and D-1 is the partial integration operator. A basis for expanding w is suggested that permits explicit integration in the expression of f. In fitting data, it is also useful to regularize f by penalizing the integral of w(2) since this is a measure of the relative curvature in f. Applications are discussed to monotone nonparametric regression, to the transformation of the dependent variable in non-linear regression and to density estimation.