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利用多元自适应回归样条进行非线性时间序列建模AI 解读

Multivariate Adaptive Regression Splines (MARS) is a new methodology, due to Friedman, for nonlinear regression modeling. MARS can be conceptualized as a generalization of recursive partitioning that uses spline fitting in lieu of other simple fitting functions. Given a set of predictor variables, MARS fits a model in the form of an expansion in product spline basis functions of predictors chosen during a forward and backward recursive partitioning strategy. MARS produces…

多元总体选择的贝叶斯与极小极大样本可容许程序AI 解读

Let P be a class of independent populations each having an underlying multivariate normal distribution with unknown mean vector. Independent vector samples of common size n are drawn from each population in P. We say that a subset G of P is δ*-Pareto-optimal (in P) if no other population in P has the property that each component of its mean vector is larger by at least δ* ≥ 0 than the corresponding component of the mean vector of any given population in G. This article…

利用傅里叶方法进行离散拉普拉斯平滑AI 解读

An approach to multidimensional smoothing is introduced that is based on a penalized likelihood with a modified discretized Laplacian penalty term. The choice of penalty simplifies computational difficulties associated with standard multidimensional Laplacian smoothing methods yet without compromising mean squared error characteristics, at least on the interior of the region of interest. For linear smoothing in hyper-rectangular domains, which has wide application in image…

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多元自适应回归样条AI 解读

A new method is presented for flexible regression modeling of high dimensional data. The model takes the form of an expansion in product spline basis functions, where the number of basis functions as well as the parameters associated with each one (product degree and knot locations) are automatically determined by the data. This procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties. Unlike recursive partitioning,…

综述：非参数密度估计的近期发展AI 解读

Advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research, and especially the development of nonparametric data analysis procedures. In particular, theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics, such as nonparametric regression, nonparametric discrimination, and nonparametric pattern recognition. This…

非参数密度估计的近期发展AI 解读

Advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research, and especially the development of nonparametric data analysis procedures. In particular, theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics, such as nonparametric regression, nonparametric discrimination, and nonparametric pattern recognition. This…

正则化信号恢复中的噪声估计AI 解读

Several estimators of variance are compared in the context of problems where smoothing is incorporated in the estimation of regression functions. Many of the estimators are scaled quadratic forms in the data, and some incorporate steps that try to accommodate substantial nonconstancy in the response function. The estimators are described and an account is provided of an empirical study based on four one-dimensional source functions of varying degrees of irregularity.

惩罚似然及相关估计量的渐近分析AI 解读

A general approach to the first order asymptotic analysis of penalized likelihood and related estimators is described. The method gives expansions for the systematic and random error. Asymptotic convergence rates in a family of spectral norms are obtained. The theory applies to a broad range of function estimation problems including nonparametric density, hazard and generalized regression curve estimation. Some examples are provided.

利用正交级数指数形式进行密度估计AI 解读

If a density is represented as an exponential of a series of orthogonal polynomials, it has a simple likelihood with a complete set of sufficient statistics. To ensure that the maximum likelihood equations are well conditioned, the polynomials should be orthogonal with respect to the estimated density. The optimal number of terms in the series can be determined by minimizing an estimate of a loss function. The penalized likelihood equations can be solved with little extra…

非高斯回归模型中的自适应样条平滑AI 解读

A convenient and efficient algorithm for penalized likelihood smoothing spline regression is proposed and illustrated. The amount of smoothing is tuned adaptively via the generalized cross-validation method. Under certain conditions, the method is shown to approximately minimize the expected Kullback-Leibler discrepancy in modeling one-parameter exponential family observations through the canonical parameter. The algorithm is applied to fit logistic models using interaction…

利用正交级数指数形式进行密度估计AI 解读

If a density is represented as an exponential of a series of orthogonal polynomials, it has a simple likelihood with a complete set of sufficient statistics. To ensure that the maximum likelihood equations are well conditioned, the polynomials should be orthogonal with respect to the estimated density. The optimal number of terms in the series can be determined by minimizing an estimate of a loss function. The penalized likelihood equations can be solved with little extra…

EM 算法在惩罚似然估计中的应用AI 解读

SUMMARY The EM algorithm is a popular approach to maximum likelihood estimation but has not been much used for penalized likelihood or maximum a posteriori estimation. This paper discusses properties of the EM algorithm in such contexts, concentrating on rates of convergence, and presents an alternative that is usually more practical and converges at least as quickly.

回归函数的估计AI 解读

In this paper, an entropy approach is proposed to establish rates of convergence for estimators of a regression function. General regression problems are considered, with linear regression, splines and isotonic regression as special cases. The estimation methods studied are least squares, least absolute deviations and penalized least squares. Common features of these methods and various regression problems are highlighted.

利用Pitman接近性准则比较线性估计量AI 解读

A method is given for the comparison of two linear forms of a common random vector under the criterion of Pitman's measure of closeness. Assuming multivariate normality of the random vector, one can determine exact expressions for the closeness probabilities. The applicability of the theory is illustrated on the comparison of ridge regression estimators.

边际尾概率近似与标量参数推断AI 解读

In many situations, inference for a scalar parameter in the presence of nuisance parameters requires integration of either a joint density of pivotal quantities or a joint posterior density. For such inference, accurate approximations of marginal tail probabilities are useful to avoid high-dimensional integrals. Two tail probability approximations are developed in this paper. Numerical results given for conditional inference in location-scale and linear regression models…

时变协变量效应的非参数生存分析：惩罚偏似然方法AI 解读

Techniques are developed for nonparametric analysis of data under a Cox-regression-like model permitting time-dependent covariate effects determined by a regression function $\beta_0(t)$. Estimators resulting from maximization of an appropriate penalized partial likelihood are shown to exist and a computational approach is outlined. Weak uniform consistency (with a rate of convergence) and pointwise asymptotic normality of the estimators are established under regularity…

高维对数线性模型最大似然估计量的 BootstrapAI 解读

The notion of a bootstrap estimator of the distribution of the maximum likelihood estimator in log-linear models is defined for common sampling models. It is shown that the bootstrap estimator is consistent under assumptions which allow the dimension of the model to increase to infinity. Such an approach allows treatment of large, sparse contingency tables.

无需数值积分的离散响应模型模拟矩估计方法AI 解读

This paper proposes a simple modification of a conventional generalized method of moments estimator for a discrete response model, replacing response probabilities that require numerical integration with estimators obtained by Monte Carlo simulation. This method of simulated moments does not require precise estimates of these probabilities, as the law of large numbers operating across observations controls simulation error, and, hence, can use simulations of practical size.…

离散含噪数据下积分方程正则化解的收敛速率AI 解读

Given data $y_i = (Kg)(u_i) + \varepsilon_i$ where the $\varepsilon$'s are random errors, the $u$'s are known, $g$ is an unknown function in a reproducing kernel space with kernel $r$ and $K$ is a known integral operator, it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points $\{u_i\}$ become dense in the interval of interest. These rates are shown to depend on the eigenvalue asymptotics of $KRK^\ast$, where $R$ is…

模式理论的进展AI 解读

Pattern theory offers concepts for modelling images and methods for making inferences from observed images. This will be described briefly and illustrated by examples. We shall present limit theorems for the Markov processes on graphs (that are basic to pattern theory) motivated by computational considerations. They will yield approximations that have been exploited to make the inference algorithms computationally feasible. We shall also consider the problem of estimating…

高维模型的递归回归及其在生长曲线与重复测量中的应用AI 解读

A fixed-effects formulation of repeated-measures and growth-curve problems usually leads to an unwieldy linear model, so mixed models are widely used for inference that the conditional linear error model could otherwise support with weaker distributional assumptions. Very high-dimensional regressions (not necessarily arising this way) can be fitted by the proposed alternating algorithm, which partitions the design matrix into singly manageable strips and recursively calls a…

线性回归中的贝叶斯变量选择AI 解读

This article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable. It is based on a Bayesian approach, intended to be as objective as possible. A probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model. The method is not fully Bayesian, however, because the ultimate…

正则化方法估计量的近似AI 解读

The Tikhonov method of regularization (MOR) estimator provides a general method for estimation of a nonparametric regression parameter in an abstract linear model with discrete noisy data. An asymptotic analysis is given in which the discrete estimation problem is approximated by a continuous one. Rates of convergence are calculated in a family of norms natural to the problem. The general theory is applied to the estimation of functions from noisy evaluations of the function…