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高维数据拟合简单模型时的 F 统计量推断AI 解读

We study linear subset regression in the context of the high-dimensional overall model $y = \vartheta +\theta ' z + \epsilon $ with univariate response y and a d -vector of random regressors z , independent of $\epsilon $ . Here, “high-dimensional” means that the number d of available explanatory variables is much larger than the number n of observations. We consider simple linear submodels where y is regressed on a set of p regressors given by $x = M'z$ , for some $d \times p$…

稳定算法的条件预测推断AI 解读

We investigate generically applicable and intuitively appealing prediction intervals based on k-fold cross-validation. We focus on the conditional coverage probability of the proposed intervals, given the observations in the training sample (hence, training conditional validity), and show that it is close to the nominal level, in an appropriate sense, provided that the underlying algorithm used for computing point predictions is sufficiently stable when feature-response…

多面体约束条件下模型选择后置信区间的长度AI 解读

Valid inference after model selection is currently a very active area of research. The polyhedral method, pioneered by Lee, et al. (2016), allows for valid inference after model selection if the model selection event can be described by polyhedral constraints. In that reference, the method is exemplified by constructing two valid confidence intervals when the Lasso estimator is used to select a model. We here study the length of these intervals. For one of these confidence…

模型选择后预测器的有效置信区间AI 解读

We consider inference post-model-selection in linear regression. In this setting, Berk et al. [Ann. Statist. 41 (2013a) 802–837] recently introduced a class of confidence sets, the so-called PoSI intervals, that cover a certain nonstandard quantity of interest with a user-specified minimal coverage probability, irrespective of the model selection procedure that is being used. In this paper, we generalize the PoSI intervals to confidence intervals for post-model-selection…

高维数据中简单模型拟合的预测表现AI 解读

We study linear subset regression in the context of a high-dimensional linear model. Consider $y=\vartheta +\theta 'z+\epsilon $ with univariate response $y$ and a $d$-vector of random regressors $z$, and a submodel where $y$ is regressed on a set of $p$ explanatory variables that are given by $x=M'z$, for some $d\times p$ matrix $M$. Here, “high-dimensional” means that the number $d$ of available explanatory variables in the overall model is much larger than the number $p$ o…

未知方差下一元或二元正态总体均值常用置信集的可容许性AI 解读

Summary In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi. This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post model selection or post shrinkage, particularly in situations where the number of parameters is larger than the sample…

高维数据低维投影的条件分布AI 解读

We study the conditional distribution of low-dimensional projections from high-dimensional data, where the conditioning is on other low-dimensional projections. To fix ideas, consider a random $d$-vector $Z$ that has a Lebesgue density and that is standardized so that $\mathbb{E} Z=0$ and $\mathbb{E} ZZ'=I_{d}$. Moreover, consider two projections defined by unit-vectors $\alpha$ and $\beta$, namely a response $y=\alpha'Z$ and an explanatory variable $x=\beta'Z$. It has long…

模型选择后的条件预测推断AI 解读

We give a finite-sample analysis of predictive inference procedures after model selection in regression with random design. The analysis is focused on a statistically challenging scenario where the number of potentially important explanatory variables can be infinite, where no regularity conditions are imposed on unknown parameters, where the number of explanatory variables in a “good” model can be of the same order as sample size and where the number of candidate models can…

能否估计模型选择后估计量的无条件分布？AI 解读

We consider the problem of estimating the unconditional distribution of a post-model-selection estimator. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model-selection criterion such as the Akaike information criterion [AIC] or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least squares or maximum likelihood), all based on the same…

稀疏估计量与 Oracle 性质：Hodges 估计量问题的重现AI 解读

We point out some pitfalls related to the concept of an oracle property as used in Fan and Li (2001, 2002, 2004) which are reminiscent of the well-known pitfalls related to Hodges’ estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to infinity whenever the loss function is…

能否估计模型选择后估计量的条件分布？AI 解读

We consider the problem of estimating the conditional distribution of a post-model-selection estimator where the conditioning is on the selected model. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion such as AIC or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based…

模型选择后置信区间的大样本最小覆盖概率AI 解读

We give a large-sample analysis of the minimal coverage probability of the usual confidence intervals for regression parameters when the underlying model is chosen by a “conservative” (or “overconsistent”) model selection procedure. We derive an upper bound for the large-sample limit minimal coverage probability of such intervals that applies to a large class of model selection procedures including the Akaike information criterion as well as various pretesting procedures.…

收缩型估计量风险与分布估计的性能限制及一般风险下界AI 解读

We consider the problem of estimating measures of precision of shrinkage-type estimators such as their risk or distribution. The notion of shrinkage-type estimators here refers to estimators such as the James–Stein estimator and Lasso-type estimators, in addition to “thresholding” estimators such as, e.g., Hodges' so-called superefficient estimator. Although the precision measures of such estimators typically can be estimated consistently, we show that they cannot be…

模型选择与推断：事实与误解AI 解读

Model selection has an important impact on subsequent inference. Ignoring the model selection step leads to invalid inference. We discuss some intricate aspects of data-driven model selection that do not seem to have been widely appreciated in the literature. We debunk some myths about model selection, in particular the myth that consistent model selection has no effect on subsequent inference asymptotically. We also discuss an “impossibility” result regarding the estimation…

模型选择后估计量的有限样本分布：一致与非一致逼近AI 解读

In Pötscher (1991, Econometric Theory 7, 163–185) the asymptotic distribution of a post-model-selection estimator, both unconditional and conditional on selecting a correct model (minimal or not), has been derived. Limitations of these results are (i) that they do not provide information on the distribution of the post-model-selection estimator conditional on selecting an incorrect model and (ii) that the quality of this asymptotic approximation to the finite-sample…

单整过程方差未必趋于无穷：平稳过程部分和的相关结果AI 解读

For a process with stationary first differences a necessary and sufficient condition for the variance of the process to be unbounded is given. An example shows that the variance of an integrated process—although unbounded—need not diverge to infinity. Sufficient conditions for the variance of an integrated process to diverge to infinity are provided.