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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.…

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…

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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…

随机过程正则化方法的极小极大性AI 解读

The idea of smoothing-spline interpolation is generalized to propose an estimator for the mean function of a stochastic process. A minimax property of the proposed estimator is then demonstrated under the usual squared loss function.

两种非参数最大惩罚似然密度估计量的一致性AI 解读

We study the consistency properties of a nonparametric estimator $f_n$ of a density function $f$ on the real line, which is known as the "first MPLE of Good and Gaskins," and which is obtained by maximizing the likelihood functional multiplied by the roughness penality $\exp\{- \alpha \int (f'/f)^2 f\}$ with $\alpha > 0$. Under modest assumptions on the density function $f$, and letting $\alpha = \alpha_n \rightarrow \infty$ and $\alpha_n/n \rightarrow 0$ a.s. as $n \rightarrow \infty$…

用最大惩罚似然方法估计概率密度函数AI 解读

A class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density. The limiting case of the estimates as the amount of smoothing increases has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty. The estimates are shown to be the solution of an unconstrained convex optimization problem,…

Maximum likelihood estimation of the parameters of the binary logistic regression model for pr(H|x) is discussed with separate discussion of sampling from (i) the conditional distribution of H given x, (ii) the joint distribution of H and x, and (iii) the conditional distribution of x given H. Difficulties associated with continuous x in the latter sampling scheme are discussed. To avoid these, penalized maximum likelihood estimates are introduced, which give estimates of…

逻辑回归与判别分析中的惩罚极大似然估计AI 解读

Maximum likelihood estimation of the parameters of the binary logistic regression model for pr(H|x) is discussed with separate discussion of sampling from (i) the conditional distribution of H given x, (ii) the joint distribution of H and x, and (iii) the conditional distribution of x given H. Difficulties associated with continuous x in the latter sampling scheme are discussed. To avoid these, penalized maximum likelihood estimates are introduced, which give estimates of…

不依赖坐标的岭回归界AI 解读

The dependence of ridge-regression estimates on the choice of parameterization is defined with reference to the hull of all possible ridge estimates for all possible parameterizations. A reanalysis of Hoerl and Kennard's (1970) 10-factor example and Marquardt and Snee's (1975) acetylene data reveals that most conclusions depend on the parameterization.

利用岭回归估计营销中的直接滞后效应AI 解读

A prediction criterion for ridge regression is developed and used for estimating lagged effect patterns by means of the direct lag model. As examples, the Lydia Pinkham monthly and annual data are analyzed, showing that lag patterns can be obtained that are in agreement across time intervals.

不依赖坐标表示的岭回归界AI 解读

The dependence of ridge-regression estimates on the choice of parameterization is defined with reference to the hull of all possible ridge estimates for all possible parameterizations. A reanalysis of Hoerl and Kennard's (1970) 10-factor example and Marquardt and Snee's (1975) acetylene data reveals that most conclusions depend on the parameterization.

均方误差的推广及其在岭回归中的应用AI 解读

There is a typographical error in equation (5.3) on p~67.The correct version of (5.3) is:This error does not affect the rest of the paper. Generalizations of Mean Square Error Applied to Ridge Regression

癌症相关非平稳泊松过程强度函数的非参数估计AI 解读

An attempt is made to model the appearance times of metastases as a nonstationary Poisson process. Three algorithms are developed for this task. The first follows the kernel approach used in probability density estimation by Parzen and Rosenblatt; the second extends the work of Grenander on mortality measurements to a more general censoring scheme appropriate for the present application; the third employs a discrete maximum penalized likelihood approach. We obtain estimates…

We demonstrate the admissible Bayes character of two residual error criteria for subset selection of independent variables in normal multivariate regression models. In particular, suppose a linear model includes the independent variable list $X$, and suppose all additional independent variable subsets of size $s$ (fixed) from list $Z$ are under consideration for inclusion in the model. Let $\mathbf{\Sigma}$ be the regression error covariance matrix and $\hat{\mathbf{\Sigma}}(\hat{\mathbf{\Sigma}}_\mathscr{J})$…

利用置信界选择岭回归的偏置参数AI 解读

Consider the multiple linear regression model Y = Xβ + ∈, ∈ ∼ Ň(0, σ2 I) where the matrix S = X' X is ill conditioned. A confidence bound approach is developed for choosing k in the ridge estimator β*(k) = (S + kI)–1 X' Y as follows: A parameter is defined that is essentially the largest (constant) k one could use and still have β*(k)'s mean squared error (MSE) be less than MSE (where is the usual estimate of β). A procedure is then developed to obtain a lower confidence…

若干岭估计量的模拟研究AI 解读

Ridge regression, which defines a class of estimators indexed by a biasing parameter k, is an alternative to least squares estimation in the multiple linear regression model. Many algorithms for the biasing parameter have been proposed in the statistical literature. The present study identifies 10 promising algorithms and systematically evaluates and compares them using Monte Carlo methods. Three algorithms perform well overall. A two-parameter estimator offers potential…