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参数维度随样本量增长时的自适应弹性网AI 解读

We consider the problem of model selection and estimation in situations where the number of parameters diverges with the sample size. When the dimension is high, an ideal method should have the oracle property (Fan and Li, 2001; Fan and Peng, 2004) which ensures the optimal large sample performance. Furthermore, the high-dimensionality often induces the collinearity problem which should be properly handled by the ideal method. Many existing variable selection methods fail to…

离散观测多维过程的共同跳跃检验AI 解读

We consider a bivariate process Xt=(Xt1, Xt2), which is observed on a finite time interval [0, T] at discrete times 0, Δn, 2Δn, …. Assuming that its two components X1 and X2 have jumps on [0, T], we derive tests to decide whether they have at least one jump occurring at the same time (“common jumps”) or not (“disjoint jumps”). There are two different tests for the two possible null hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level,…

Lasso 与 Dantzig selector 的统一分析AI 解读

We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the ℓp estimation loss for 1≤p≤2 in the linear model when the number of variables can be much larger than the sample size.

在线预测线性回归AI 解读

We consider the on-line predictive version of the standard problem of linear regression; the goal is to predict each consecutive response given the corresponding explanatory variables and all the previous observations. The standard treatment of prediction in linear regression analysis has two drawbacks: (1) the classical prediction intervals guarantee that the probability of error is equal to the nominal significance level ɛ, but this property per se does not imply that the…

通过熵惩罚在凸包中实现稀疏恢复AI 解读

Let (X, Y) be a random couple in S×T with unknown distribution P and (X1, Y1), …, (Xn, Yn) be i.i.d. copies of (X, Y). Denote Pn the empirical distribution of (X1, Y1), …, (Xn, Yn). Let h1, …, hN: S↦[−1, 1] be a dictionary that consists of N functions. For λ∈ℝN, denote fλ:=∑j=1Nλjhj. Let ℓ: T×ℝ↦ℝ be a given loss function and suppose it is convex with respect to the second variable. Let (ℓ•f)(x, y):=ℓ(y; f(x)). Finally, let Λ⊂ℝN be the simplex of all probability distributions…

一般半参数共享脆弱性模型下的病例对照生存分析：伪完整似然方法AI 解读

In this work we deal with correlated failure time (age at onset) data arising from population-based, case-control studies, where case and control probands are selected by population-based sampling and an array of risk factor measures is collected for both cases and controls and their relatives. Parameters of interest are effects of risk factors on the failure time hazard function and within-family dependencies among failure times after adjusting for the risk factors. Due to…

Logistic 项目反应模型的非线性序贯设计及计算机自适应测验应用AI 解读

Computerized adaptive testing is becoming increasingly popular due to advancement of modern computer technology. It differs from the conventional standardized testing in that the selection of test items is tailored to individual examinee’s ability level. Arising from this selection strategy is a nonlinear sequential design problem. We study, in this paper, the sequential design problem in the context of the logistic item response theory models. We show that the adaptive…

This paper offers a new approach to modeling and forecasting of nonstationary time series with applications to volatility modeling for financial data. The approach is based on the assumption of local homogeneity: for every time point, there exists a historical interval of homogeneity, in which the volatility parameter can be well approximated by a constant. The proposed procedure recovers this interval from the data using the local change point (LCP) analysis. Afterward, the…

复合函数的非参数估计AI 解读

We study the problem of nonparametric estimation of a multivariate function g: ℝd→ℝ that can be represented as a composition of two unknown smooth functions f: ℝ→ℝ and G: ℝd→ℝ. We suppose that f and G belong to known smoothness classes of functions, with smoothness γ and β, respectively. We obtain the full description of minimax rates of estimation of g in terms of γ and β, and propose rate-optimal estimators for the sup-norm loss. For the construction of such estimators, we…

对数凹密度最大似然估计的极限分布理论AI 解读

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, i.e. a density of the form f(0) = exp varphi(0) where varphi(0) is a concave function on R. Existence, form, characterizations and uniform rates of convergence of the MLE are given by Rufibach (2006) and Dümbgen and Rufibach (2007). The characterization of the log-concave MLE in terms of distribution functions is the same (up to sign) as the characterization of…

主成分限制最大似然估计量的一致性AI 解读

In this paper we consider two closely related problems: estimation of eigenvalues and eigenfunctions of the covariance kernel of functional data based on (possibly) irregular measurements, and the problem of estimating the eigenvalues and eigenvectors of the covariance matrix for high-dimensional Gaussian vectors. In [A geometric approach to maximum likelihood estimation of covariance kernel from sparse irregular longitudinal data (2007)], a restricted maximum likelihood…

拓展经验似然的适用范围AI 解读

This article extends the scope of empirical likelihood methodology in three directions: to allow for plug-in estimates of nuisance parameters in estimating equations, slower than $\sqrt{n}$-rates of convergence, and settings in which there are a relatively large number of estimating equations compared to the sample size. Calibrating empirical likelihood confidence regions with plug-in is sometimes intractable due to the complexity of the asymptotics, so we introduce a…

贝叶斯变量选择程序的一致性AI 解读

It has long been known that for the comparison of pairwise nested models, a decision based on the Bayes factor produces a consistent model selector (in the frequentist sense). Here we go beyond the usual consistency for nested pairwise models, and show that for a wide class of prior distributions, including intrinsic priors, the corresponding Bayesian procedure for variable selection in normal regression is consistent in the entire class of normal linear models. We find that…

中值滤波是否真的比线性滤波更能保留边缘？AI 解读

Image processing researchers commonly assert that “median filtering is better than linear filtering for removing noise in the presence of edges.” Using a straightforward large-n decision-theory framework, this folk-theorem is seen to be false in general. We show that median filtering and linear filtering have similar asymptotic worst-case mean-squared error (MSE) when the signal-to-noise ratio (SNR) is of order 1, which corresponds to the case of constant per-pixel noise…

球面 needlet 小波的渐近理论AI 解读

We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasi-exponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial…

面板计数数据的多样本非参数检验新方法AI 解读

This paper considers the problem of multi-sample nonparametric comparison of counting processes with panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. For the problem considered, we construct two new classes of nonparametric test statistics based on the accumulated weighted differences between the rates of increase of the estimated mean functions of…

非椭圆分布预测变量的降维AI 解读

Sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor, or, when such conditions are not satisfied, rely on marginal transformation or reweighting to fulfill them approximately. For example, a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distribution. In this paper, we reformulate the commonly used dimension reduction methods, via the notion of…

双色随机强化 urn 生成的反应自适应设计：渐近性质AI 解读

This paper illustrates asymptotic properties for a response-adaptive design generated by a two-color, randomly reinforced urn model. The design considered is optimal in the sense that it assigns patients to the best treatment, with probability converging to one. An approach to show the joint asymptotic normality of the estimators of the mean responses to the treatments is provided in spite of the fact that allocation proportions converge to zero and one. Results on the rate…

有向无环图的矩阵表示与条件独立性AI 解读

For a directed acyclic graph, there are two known criteria to decide whether any specific conditional independence statement is implied for all distributions factorized according to the given graph. Both criteria are based on special types of path in graphs. They are called separation criteria because independence holds whenever the conditioning set is a separating set in a graph theoretical sense. We introduce and discuss an alternative approach using binary matrix…

独立性条件下具有 FDR 控制保证的自适应逐步下降程序AI 解读

In this work we study an adaptive step-down procedure for testing m hypotheses. It stems from the repeated use of the false discovery rate controlling the linear step-up procedure (sometimes called BH), and makes use of the critical constants iq/[(m+1−i(1−q)], i=1, …, m. Motivated by its success as a model selection procedure, as well as by its asymptotic optimality, we are interested in its false discovery rate (FDR) controlling properties for a finite number of hypotheses.…

从含加性噪声的数据估计凹分布函数AI 解读

We consider two nonparametric procedures for estimating a concave distribution function based on data corrupted with additive noise generated by a bounded decreasing density on (0, ∞). For the maximum likelihood (ML) estimator and least squares (LS) estimator, we state qualitative properties, prove consistency and propose a computational algorithm. For the LS estimator and its derivative, we also derive the pointwise asymptotic distribution. Moreover, the rate n−2/5 achieved…

未知方差下的高斯模型选择AI 解读

Let Y be a Gaussian vector whose components are independent with a common unknown variance. We consider the problem of estimating the mean μ of Y by model selection. More precisely, we start with a collection $\mathcal{S}=\{S_{m},m\in\mathcal{M}\}$ of linear subspaces of ℝn and associate to each of these the least-squares estimator of μ on Sm. Then, we use a data driven penalized criterion in order to select one estimator among these. Our first objective is to analyze the…

似然比检验与奇异点AI 解读

Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff’s theorem hold for semi-algebraic sets such that the asymptotics are determined by the…

错误发现率与渐近最优拒绝曲线AI 解读

In this paper we introduce and investigate a new rejection curve for asymptotic control of the false discovery rate (FDR) in multiple hypotheses testing problems. We first give a heuristic motivation for this new curve and propose some procedures related to it. Then we introduce a set of possible assumptions and give a unifying short proof of FDR control for procedures based on Simes’ critical values, whereby certain types of dependency are allowed. This methodology of proof…