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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…

实例回归分析AI 解读

Simple Linear Regression Multiple Linear Regression Regression Diagnostics: Detection of Model Violations Qualitative Variables as Predictors Transformation of Variables Weighted Least Squares The Problem of Correlated Errors Analysis of Collinear Data Biased Estimation of Regression Coefficients Variable Selection Procedures Logistic Regression Appendix References Index.

岭回归估计的极小极大性质AI 解读

The technique of ridge regression, first proposed by Hoerl and Kennard, has become a popular tool for data analysts faced with a high degree of multicollinearity in their data. By using a ridge estimator, one hopes to both stabilize one's estimates (lower the condition number of the design matrix) and improve upon the squared error loss of the least squares estimator. Recently, much attention has been focused on the latter objective. Building on the work of Stein and others,…

一类岭回归估计量的频率学派分析AI 解读

We discuss ridge regression estimators as a class, showing how six recently proposed estimators can thus be viewed, and give reasons for preferring one of these (empirical Bayes maximum likelihood estimator) to the other five. Results are based on analysis rather than simulations and on a frequentist rather than Bayesian viewpoint.

一类岭回归估计量的频率学派分析AI 解读

We discuss ridge regression estimators as a class, showing how six recently proposed estimators can thus be viewed, and give reasons for preferring one of these (empirical Bayes maximum likelihood estimator) to the other five. Results are based on analysis rather than simulations and on a frequentist rather than Bayesian viewpoint.

基于离散最大惩罚似然准则的非参数密度估计AI 解读

A nonparametric probability density estimator is proposed that is optimal with respect to a discretized form of a continuous penalized-likelihood criterion functional. Approximation results relating the discrete estimator to the estimate obtained by solving the corresponding infinite-dimensional problem are presented. The discrete estimator is shown to be consistent. The numerical implementation of this discrete estimator is outlined and examples displayed. A simulation…

对若干岭回归方法的批评AI 解读

Ridge estimates seem motivated by a belief that least squares estimates tend to be too large, particularly when there is multicollinearity. The ridge solution is to supplement the data by stochastically shrinking the estimates toward zero. Although flexibility is provided by the abstention from exact exclusion restrictions, ridge regression retains many weaknesses of similarly motivated procedures : a neglect of the basic fact that linear transformations should not change…

惩罚似然密度估计与峰值识别：散射及陨石数据示例AI 解读

The (maximum) penalized-likelihood method of probability density estimation and bump-hunting is improved and exemplified by applications to scattering and chondrite data. We show how the hyperparameter in the method can be satisfactorily estimated by using statistics of goodness of fit. A Fourier expansion is found to be usually more expeditious than a Hermite expansion but a compromise is useful. The best fit to the scattering data has 13 bumps, all of which are evaluated…

自适应多元岭回归AI 解读

A multivariate version of the Hoerl-Kennard ridge regression rule is introduced. The choice from among a large class of possible generalizations is guided by Bayesian considerations; the result is implicitly in the work of Lindley and Smith although not actually derived there. The proposed rule, in a variety of equivalent forms is discussed and the choice of its ridge matrix considered. As well, adaptive multivariate ridge rules and closely related empirical Bayes procedures…

正态离散矩阵的贝叶斯推断及其在随机多元回归中的应用AI 解读

Summary This paper discusses Bayesian inference procedures for a normal dispersion matrix. Structural information for the prior mean of the dispersion matrix is incorporated into the analysis through a Normal-Wishart prior distribution. Many of the resulting Bayes estimates are invariant, consistent and asymptotically efficient. Using this procedure, a coherent Bayesian argument for stochastic multiple regression analysis is developed, where dependent and independent…

一类收缩估计量AI 解读

Summary In this paper we define a class of shrinkage estimators, all of whose members have a mean square error matrix which is less than that of the ordinary least squares estimator by a positive semidefinite matrix if (β-b *)T X T X (β-b *) ≤ σ2. The ridge regression, principal component and minimum conditional mean square error estimators are members of this class.

极小极大自适应广义岭回归估计量AI 解读

We consider the problem of estimating the vector of regression coefficients of a linear model using generalized ridge regression estimators where the ridge constant is chosen on the basis of the data. For general quadratic loss we produce such estimators whose risk function dominates that of the least squares procedure provided the number of regressors is at least three. We study the problem by the usual reduction to estimating the mean vector of a multivariate normal…