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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…

广义线性模型中回归函数的自动平滑AI 解读

We consider the penalized likelihood method for estimating nonparametric regression functions in generalized linear models (Nelder and Wedderburn 1972) and present a generalized cross-validation procedure for empirically assessing an appropriate amount of smoothing in these estimates. Asymptotic arguments and numerical simulations are used to show that the generalized cross-validatory procedure preforms well from the point of view of a weighted mean squared error criterion.…

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从 Stein 无偏风险估计到广义交叉验证AI 解读

This paper concerns the method of generalized cross validation (GCV), a promising way of choosing between linear estimates. Based on Stein estimates and the associated unbiased risk estimates (Stein, 1981), a new approach to GCV is developed. Many consistency results are obtained for the cross-validated (Steinized) estimates in the contexts of nearest-neighbor nonparametric regression, model selection, ridge regression, and smoothing splines. Moreover, the associated Stein's…

条件数与极小极大岭回归估计量AI 解读

Ridge regression was originally formulated with two goals in mind: improvement in mean squared error and numerical stability of the coefficient estimates. Conditions are given under which a minimax ridge regression estimator can also improve numerical stability, a quantity that can be measured with the condition number of the matrix to be inverted. The consequences of trading numerical stability for minimaxity are also discussed.

一类非参数密度与回归估计量AI 解读

A class of maximum penalized likelihood estimators (MPLE) of the density function $f$ is constructed, through the use of a rather general roughness-penalty functional. This class contains all the density estimates in the literature that arise as solutions to MPLE problems with penalties on $f^{1/2}$. In addition, the flexibility of the penalty functional permits the construction of new spline estimates with improved performance at the peaks and valleys of the density curves.…

样条平滑：等价变带宽核方法AI 解读

The spline smoothing approach to nonparametric regression and curve estimation is considered. It is shown that, in a certain sense, spline smoothing corresponds approximately to smoothing by a kernel method with bandwidth depending on the local density of design points. Some exact calculations demonstrate that the approximation is extremely close in practice. Consideration of kernel smoothing methods demonstrates that the way in which the effective local bandwidth behaves in…

图形投影寻踪的渐近理论AI 解读

Mathematical tools are developed for describing low-dimensional projections of high-dimensional data. Theorems are given to show that under suitable conditions, most projections are approximately Gaussian.

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用普通最小二乘进行 Logistic 回归与判别分析AI 解读

If the observations for fitting a polytomous logistic regression model satisfy certain normality assumptions, the maximum likelihood estimates of the regression coefficients are the discriminant function estimates. This article shows that these estimates, their unbiased counterparts, and associated test statistics for variable selection can be calculated using ordinary least squares regression techniques, thereby providing a convenient method for fitting logistic regression…

回归中一步变量选择的最优性质AI 解读

We consider the selection of the best subset of independent variables of a fixed size for possible inclusion in a regression model. The classical procedures (largest $\mathbb{R}^2$ to enter) are shown to be uniformly invariant Bayes in the sense of Paulson (1952) and Kudo (1956).

平滑样条：回归、导数与反卷积AI 解读

The statistical properties of a cubic smoothing spline and its derivative are analyzed. It is shown that unless unnatural boundary conditions hold, the integrated squared bias is dominated by local effects near the boundary. Similar effects are shown to occur in the regularized solution of a translation-kernel integral equation. These results are derived by developing a Fourier representation for a smoothing spline.

公共项目分析：一种新的分类数据方法AI 解读

I: Foundations.- 1. Multidimensional Contingency Tables.- 2. Guidelines for Analysis.- What Are Categorical Data?.- Is This a Categorical Data Problem?.- What Function Should Be Analyzed?.- What Sample Size Is Required?.- Are Zero Cells a Problem?.- How Many Analyses Should Be Performed?.- What Levels of Significance Should Be used?.- Summary.- 3. Contingency Table Analysis: The WLS Approach.- The Contingency Table.- Formation of Functions.- The Variance of a Function.- The…