论文检索

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.

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…

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…

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…

多元岭回归AI 解读

A multivariate linear regression model with q responses as a linear function of p independent variables is considered with a p × q parameter matrix B. The least-squares or normal-theory maximum likelihood estimate of B is deficient in that it takes no account of the ‘across regression’ correlations, and ignores the Stein effect. A remedy was offered by Brown & Zidek (1980) in the form of a multivariate ridge estimator. A richer class of estimators is obtained here by casting…

通过非参数估计构造的一类线性回归参数估计量AI 解读

Given a $(p + 1)$-dimensional random vector $(X, Y)$ where $f$ is the unknown density of $X$, the parameters of the multiple linear regression function $\alpha(x) = E(Y/X = x) = x\beta$ may be estimated from a sample $\{(X_1, Y_1), \cdots, (X_n, Y_n)\}$ by minimizing the functional $\hat{\psi}(\beta) = \int(\hat{\alpha}_n(x) - x\beta)^2\hat{f}_n(x) dx$, where $\hat{\alpha}_n$ and $\hat{f}_n$ may be any of a large class of nonparametric estimators of $\alpha$ and $f$. The…

分支曲线的非参数估计AI 解读

The spline smoothing approach to nonparametric regression is extended to cover the estimation of curves that branch at certain specified points in time. This extension is motivated and illustrated by data from an experiment on sunflowers. Apart from providing insight into this particular data-analytic problem, the work described demonstrates the power of the roughness penalty approach in coping with curve estimation of a nonstandard kind. Particularly in the context of…

大型模型族的快速模型选择程序AI 解读

An efficient procedure for model selection from large families of models is described. It is closely related to the all possible models approach but is considerably faster. It is based on two principles: first, if a model is accepted, then all models that include it are considered to be accepted; second, if a model is rejected, then all of its submodels are considered to be rejected. Application of the procedure to variable selection in multiple regression is illustrated.…

风险率与强度函数的惩罚似然估计AI 解读

SUMMARY A nonparametric estimate is proposed for the hazard function using penalized likelihood methods. The estimate turns out to be a hyperbolic spline function which is continuous with its first derivative discontinuous only at a finite number of points. It always assumes positive values, unlike Anderson and Senthilselvan’s (1980) quadratic spline estimate. This method is modified for the estimation of the intensity function of a nonstationary Poisson process. An example…

关于多元岭回归AI 解读

A multivariate linear regression model with q responses as a linear function of p independent variables is considered with a p × q parameter matrix B. The least-squares or normal-theory maximum likelihood estimate of B is deficient in that it takes no account of the ‘across regression’ correlations, and ignores the Stein effect. A remedy was offered by Brown & Zidek (1980) in the form of a multivariate ridge estimator. A richer class of estimators is obtained here by casting…

岭回归中 Mallows 准则与广义交叉验证的渐近最优性及样条应用AI 解读

The asymptotic optimality of Mallows' $C_L$ and generalized cross-validation is demonstrated in the setting of ridge regression. An application is made to spline smoothing in nonparametric regression. A counterexample is given to help understand why sometimes GCV may not be asymptotically optimal. The coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of GCV. The proff is based on the connection between…

图像恢复中的若干平滑技术AI 解读

SUMMARY The problem is considered of restoring a blurred and/or noisy image using various regularization prescriptions. Preliminary work concerns the invertibility of point spread functions and the construction of a stochastic model for images. A simple linear regularization procedure is introduced, as well as the special cases of constrained deconvolution, maximum entropy restoration and least-squares filtering. Optimal choices for the degree of smoothing are obtained for…

异方差判别分析中的变量选择AI 解读

The likelihood ratio test statistic for the identity in means and covariance matrices of k normal populations has a well-known step-down decomposition measuring the contribution of each component of the vector observation. This decomposition in turn gives rise to three components testing the residual homo-scedasticity of each variable, the parallelism of its regression on its predecessors, and the identity of location. A variety of uses of this decomposition in selecting…

关于平滑先验与非线性回归的一个注记AI 解读

A generalization of a nonparametric estimator proposed by Shiller (1984) is considered. The estimator is shown to be directly related to penalized least squares estimation and spline smoothing. Some simplifications concerning the computation of Shiller's estimator and corresponding posterior covariances are indicated.

岭回归中自助法与交叉验证的应用AI 解读

Several existing methods for the choice of the ridge parameter are reviewed, and a bootstrap method is proposed. The bootstrap provides independent measures of prediction errors based on multiple predictions along with an estimate of the standard error of prediction. The bootstrap and selected competitors are compared through Monte Carlo simulations for various degrees of design matrix collinearity and varying levels of signal-to-noise ratio. The procedure is also…