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We extend Robins’ theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments. In particular we establish versions of the key results of the discrete theory: the $g$-computation formula and a collection of powerful characterizations of the $g$-null hypothesis of no treatment effect. This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome…

布朗运动最小凹上包络的边际密度AI 解读

A clean, closed form, joint density is derived for Brownian motion, its least concave majorant, and its derivative, all at the same fixed point. Some remarkable conditional and marginal distributions follow from this joint density. For example, it is shown that the height of the least concave majorant of Brownian motion at a fixed time point has the same distribution as the distance from the Brownian motion path to its least concave majorant at the same fixed time point.…

单调函数的似然比检验AI 解读

We study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function.The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data)and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion $+t^2$…

凸函数估计：估计量刻画与渐近理论AI 解读

We study nonparametric estimation of convexregression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case).We provide characterizations of these estimators, prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated. The asymptotic distribution theory relies on the existence of an “invelope function” for…

凸函数估计的典范过程：积分布朗运动加四次漂移的“内包络”AI 解读

A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively. We call this process “the invelope” and show that it is an almost surely uniquely defined function of integrated Brownian motion. Its role is comparable to the role of the greatest convex minorant of Brownian motion plus a parabolic drift in the problem…

结构自适应降维方法AI 解读

We propose a new method of effective dimension reduction for a multi-index model which is based on iterative improvement of the family of average derivative estimates. The procedure is computationally straightforward and does not require any prior information about the structure of the underlying model. We show that in the case when the effective dimension $m$ of the index space does not exceed 3, this space can be estimated with the rate $n^{-1/2}$ under rather mild…

生存时间的广义加性回归模型AI 解读

We present a non-parametric survival model with two time-scales. The time-scales are equivalent up to a constant that varies over the subjects. Covariate effects are modelled linearly on each time scale by additive Aalen models. Estimators of the cumulative intensities on the two time-scales are suggested by solving approximate local maximum likelihood estimating equations. The local estimating equations necessitate only the choice of one bandwidth. The estimators are…

连续时间半鞅回归模型中的参数序贯估计AI 解读

We consider the problem of parameter estimation for multidimensional continuous-time linear stochastic regression models with an arbitrary finite number of unknown parameters and with martingale noise. The main result of the paper claims that the unknown parameters can be estimated with prescribed mean-square precision in this general model providing a unified description of both discrete and continuous time process. Among the conditions on the regressors there is one…

基于自助法的非参数回归显著性检验AI 解读

This paper proposes a test for selecting explanatory variables in nonparametric regression. The test does not need to estimate the conditional expectation function given all the variables, but only those which are significant under the null hypothesis. This feature is computationally convenient and solves, in part, the problem of the “curse of dimensionality” when selecting regressors in a nonparametric context. The proposed test statistic is based on functionals of a $U$-pro…

依赖训练序列下的最近邻分类AI 解读

The asymptotic classification risk for nearest neighbor procedures is well understood in the case of i.i.d. training sequences. In this article, we generalize these results to a class of dependent models including hidden Markov models. In the case where the observed patterns have Lebesgue densities, the asymptotic risk takes the same expression as in the i.i.d. case. For discrete distributions, we show that the asymptotic risk depends on the rule used for breaking ties of…

极值分布谱测度的非参数估计AI 解读

Let $(\mathcal{X}_1, \mathcal{Y}_1),\dots,(\mathcal{X}_n, \mathcal{Y}_n)$ be a random sample from a bivariate distribution function $F$ in the domain of max-attraction of a distribution function $G$. This $G$ is characterised by the two extreme value indices and its spectral or angular measure. The extreme value indices determine both the marginals and the spectral measure determines the dependence structure of $G$. One of the main issues in multivariate extreme value theory…

最小非参数容忍区域AI 解读

We present a new, natural way to construct nonparametric multivariate tolerance regions. Unlike the classical nonparametric tolerance intervals, where the endpoints are determined by beforehand chosen order statistics, we take the shortest interval that contains a certain number of observations. We extend this idea to higher dimensions by replacing the class of intervals by a general class of indexing sets, which specializes to the classes of ellipsoids, hyperrectangles or…

半参数模型的多尺度最大似然分析及应用AI 解读

A special semiparametric model for a univariate density is introduced that allows analyzing a number of problems via appropriate transformations. Two problems treated in some detail are testing for the presence of a mixture and detecting a wear-out trend in a failure rate. The analysis of the semiparametric model leads to an approach that advances the maximum likelihood theory of the Grenander estimator to a multiscale analysis. The construction of the corresponding test…

混合模型复杂度的一致估计AI 解读

The consistent estimation of mixture complexity is of fundamental importance in many applications of finite mixture models. An enormous body of literature exists regarding the application, computational issues and theoretical aspects of mixture models when the number of components is known, but estimating the unknown number of components remains an area of intense research effort. This article presents a semiparametric methodology yielding almost sure convergence of the…

正态混合密度极大似然与贝叶斯估计的熵及收敛速度AI 解读

We study the rates of convergence of the maximum likelihood estimator (MLE) and posterior distribution in density estimation problems, where the densities are location or location-scale mixtures of normal distributions with the scale parameter lying between two positive numbers. The true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having sub-Gaussian tails. We obtain bounds for Hellinger bracketing entropies…

非参数协方差分析AI 解读

In the problem of testing the equalityof k regression curves from independent samples, we discuss three methods using nonparametric estimators of the regression function. The first test is based on a linear combination of estimators for the integrated variance function in the individual samples and in the combined sample. The second approach transfers the classical one-way analysis of variance to the situation of comparing non-parametric curves, while the third test compares…

We examine the way in which empirical bandwidth choice affects distributional properties of nonparametric density estimators. Two bandwidth selection methods are considered in detail: local and global plug-in rules. Particular attention is focussed on whether the accuracy of distributional bootstrap approximations is appreciably influenced by using the resample version $\hat{h}*$,rather than the sample version $\hat{h}$, of an empirical bandwidth. It is shown theoretically…

Bernstein 多项式密度估计的收敛速度AI 解读

Mixture models for density estimation provide a very useful set up for the Bayesian or the maximum likelihood approach.For a density on the unit interval, mixtures of beta densities form a flexible model. The class of Bernstein densities is a much smaller subclass of the beta mixtures defined by Bernstein polynomials, which can approximate any continuous density. A Bernstein polynomial prior is obtained by putting a prior distribution on the class of Bernstein densities. The…

贪婪函数逼近：梯度提升机AI 解读

Function estimation/approximation is viewed from the perspective of numerical optimization in function space, rather than parameter space. A connection is made between stagewise additive expansions and steepest-descent minimization. A general gradient descent “boosting” paradigm is developed for additive expansions based on any fitting criterion.Specific algorithms are presented for least-squares, least absolute deviation, and Huber-M loss functions for regression, and…

通过最大化最低 D 与 D₁ 效率构造多项式回归稳健设计AI 解读

In the common polynomial regression of degree m we determine the design which maximizes the minimum of the $D$-efficiency in the model of degree $m$ and the $D_1$-efficiencies in the models of degree $m-j,\dots, m +k$ ($j, k\ge 0$ given). The resulting designs allow an efficient estimation of the parameters in the chosen regression and have reasonable efficiencies for checking the goodness-of-fit of the assumed model of degree $m$ by testing the highest coefficients in the…

用于混合模型诊断的拟合优度检验AI 解读

A simple goodness of fit test is proposed for checking distributional assumptions involved in a mixed linear model. An estimated critical value of the test statistic is derived, and is shown to be asymptotically correct under mild conditions. As a special case,the test may be applied to linear regression models to formally check distribution of the errors. Finite sample performance of the proposed test is examined and compared with that of a previously proposed test by…

二项模型的稳健拟合AI 解读

We consider the problem of robust inference for the binomial $(m, \pi)$ model. The discreteness of the data and the fact that the parameter and sample spaces are bounded mean that standard robustness theory gives surprising results. For example, the maximum likelihood estimator (MLE) is quite robust, it cannot be improved on for $m=1$ but can be for $m>1$. We discuss four other classes of estimators: $M$-estimators, minimum disparity estimators, optimal MGP estimators, and a…

广义矩条件下计算Γ极小极大决策规则的算法AI 解读

We present an algorithm for calculating a $\Gamma$-minimax decision rule, when is given by a finite number of generalized moment conditions. Such a decision rule minimizes the maximum of the integrals of the risk function with respect to all distributions in $\Gamma$. The inner maximization problem is approximated by a sequence of linear programs. This approximation is combined with an elimination technique which quickly reduces the domain of the variables of the outer…

正交饱和设计自适应分析中的错误率控制AI 解读

Individual and simultaneous confidence intervals using the data adaptively are constructed for the effects in orthogonal saturated designs under the assumption of effect sparsity. The minimum coverage probabilities of the intervals are equal to the nominal level $1 - \alpha$.