This paper develops a general theory on rates of convergence of penalized spline estimators for function estimation when the likelihood functional is concave in candidate functions, where the likelihood is interpreted in a broad sense that includes conditional likelihood, quasi-likelihood, and pseudo-likelihood. The theory allows all feasible combinations of the spline degree, the penalty order, and the smoothness of the unknown functions. According to this theory, the asymptotic behaviors of the penalized spline estimators depends on interplay between the spline knot number and the penalty parameter. The general theory is applied to obtain results in a variety of contexts, including regression, generalized regression such as logistic regression and Poisson regression, density estimation, conditional hazard function estimation for censored data, quantile regression, diffusion function estimation for a diffusion type process, and estimation of spectral density function of a stationary time series. For multi-dimensional function estimation, the theory (presented in the Supplementary Material) covers both penalized tensor product splines and penalized bivariate splines on triangulations.

翻译：本文发展了一种一般理论, 说明在可能的功能在候选函数中混为一谈时, 功能性的可能性在候选函数中, 对可能性的解释范围很广, 包括有条件的可能性、 准相似性、 假相似性。 理论允许将所有可能的样度、 惩罚顺序和未知函数的平滑性组合在一起。 根据这个理论, 受处罚的样条估计器的无症状行为取决于 样条纹结数和惩罚参数之间的相互作用。 一般理论用于在多种情况下取得结果, 包括回归、 普遍回归, 如物流回归和 Poisson回归、 密度估计、 受审查的数据的有条件危险函数估计、 微量回归、 扩散型进程的扩散函数估计, 以及固定时间序列的光谱密度函数估计。 对于多维函数估计, 理论( 在补充材料中呈现 ) 涵盖受处罚的 色条纹 和 三角形 上受处罚的双变量 。