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Many statistical procedures, including goodness-of-fit tests and methods for independent component analysis, rely critically on the estimation of the entropy of a distribution. In this paper, we seek entropy estimators that are efficient and achieve the local asymptotic minimax lower bound with respect to squared error loss. To this end, we study weighted averages of the estimators originally proposed by Kozachenko and Leonenko [Probl. Inform. Transm. 23 (1987), 95–101],…

The log-concave maximum likelihood estimator of a density on the real line based on a sample of size $n$ is known to attain the minimax optimal rate of convergence of $O(n^{-4/5})$ with respect to, for example, squared Hellinger distance. In this paper, we show that it also enjoys attractive adaptation properties, in the sense that it achieves a faster rate of convergence when the logarithm of the true density is $k$-affine (i.e., made up of $k$ affine pieces), or close to $k$…

基于稀疏投影的高维变点估计AI 解读

Summary Change points are a very common feature of ‘big data’ that arrive in the form of a data stream. We study high dimensional time series in which, at certain time points, the mean structure changes in a sparse subset of the co-ordinates. The challenge is to borrow strength across the co-ordinates to detect smaller changes than could be observed in any individual component series. We propose a two-stage procedure called inspect for estimation of the change points: first,…

随机投影集成分类AI 解读

Summary We introduce a very general method for high dimensional classification, based on careful combination of the results of applying an arbitrary base classifier to random projections of the feature vectors into a lower dimensional space. In one special case that we study in detail, the random projections are divided into disjoint groups, and within each group we select the projection yielding the smallest estimate of the test error. Our random-projection ensemble…

对数凹密度估计的全局收敛速率AI 解读

The estimation of a log-concave density on $\mathbb{R}^{d}$ represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size $n$ can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk…

Peter Hall 在高维数据与分类研究中的工作AI 解读

In this article, I summarise Peter Hall’s contributions to high-dimensional data, including their geometric representations and variable selection methods based on ranking. I also discuss his work on classification problems, concluding with some personal reflections on my own interactions with him. This article complements [Ann. Statist. 44 (2016) 1821–1836; Ann. Statist. 44 (2016) 1837–1853; Ann. Statist. 44 (2016) 1854–1866 and Ann. Statist. 44 (2016) 1867–1887], which…

稀疏主成分估计中的统计与计算权衡AI 解读

In recent years, sparse principal component analysis has emerged as an extremely popular dimension reduction technique for high-dimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse. An impressive range of estimators have been proposed; some of these are fast to compute, while others are known to achieve the minimax optimal rate over…

带形状约束的广义加性模型与指数模型AI 解读

Summary We study generalized additive models, with shape restrictions (e.g. monotonicity, convexity and concavity) imposed on each component of the additive prediction function. We show that this framework facilitates a non-parametric estimator of each additive component, obtained by maximizing the likelihood. The procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals. More generally, our methodology can be…

带错误控制的变量选择：再论稳定性选择AI 解读

Summary Stability selection was recently introduced by Meinshausen and Bühlmann as a very general technique designed to improve the performance of a variable selection algorithm. It is based on aggregating the results of applying a selection procedure to subsamples of the data. We introduce a variant, called complementary pairs stability selection, and derive bounds both on the expected number of variables included by complementary pairs stability selection that have low…

基于非参数极大似然估计的独立成分分析AI 解读

Independent Component Analysis (ICA) models are very popular semiparametric models in which we observe independent copies of a random vector $X=AS$, where $A$ is a non-singular matrix and $S$ has independent components. We propose a new way of estimating the unmixing matrix $W=A^{-1}$ and the marginal distributions of the components of $S$ using nonparametric maximum likelihood. Specifically, we study the projection of the empirical distribution onto the subset of ICA…

最近邻分类中的邻居阶数选择AI 解读

The kth-nearest neighbor rule is arguably the simplest and most intuitively appealing nonparametric classification procedure. However, application of this method is inhibited by lack of knowledge about its properties, in particular, about the manner in which it is influenced by the value of k; and by the absence of techniques for empirical choice of k. In the present paper we detail the way in which the value of k determines the misclassification error. We consider two…

袋装最近邻分类器的性质AI 解读

Summary It is shown that bagging, a computationally intensive method, asymptotically improves the performance of nearest neighbour classifiers provided that the resample size is less than 69% of the actual sample size, in the case of with-replacement bagging, or less than 50% of the sample size, for without-replacement bagging. However, for larger sampling fractions there is no asymptotic difference between the risk of the regular nearest neighbour classifier and its bagged…