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删失寿命的非参数二元密度估计AI 解读

It is well known that estimation of a bivariate cumulative distribution function of a pair of right censored lifetimes presents challenges unparalleled to the univariate case where a product-limit Kaplan–Meyer’s methodology typically yields optimal estimation, and the literature on optimal estimation of the joint probability density is next to none. The paper, for the first time in the survival analysis literature, develops the theory and methodology of sharp minimax and…

有无缺失时当前状态删失下的精确极小极大分布估计AI 解读

Nonparametric estimation of the cumulative distribution function and the probability density of a lifetime X modified by a current status censoring (CSC), including cases of right and left missing data, is a classical ill-posed problem with biased data. The biased nature of CSC data may preclude us from consistent estimation unless the biasing function is known or may be estimated, and its ill-posed nature slows down rates of convergence. Under a traditionally studied CSC,…

尺度依赖辅助变量的非参数回归AI 解读

The paper is devoted to the problem of estimation of a univariate component in a heteroscedastic nonparametric multiple regression under the mean integrated squared error (MISE) criteria. The aim is to understand how the scale function should be used for estimation of the univariate component. It is known that the scale function does not affect the rate of the MISE convergence, and as a result sharp constants are explored. The paper begins with developing a sharp-minimax…

预测变量随机缺失下的非参数回归AI 解读

Nonparametric regression with predictors missing at random (MAR), where the probability of missing depends only on observed variables, is considered. Univariate predictor is the primary case of interest. A new adaptive orthogonal series estimator is developed. Large sample theory shows that the estimator is rate-minimax and it is also sharp-minimax whenever predictors are missing completely at random (MCAR). Furthermore, confidence bands, estimation of nuisance functions,…

条件密度估计中的降维与自适应AI 解读

An orthogonal series estimator of the conditional density of a response given a vector of continuous and ordinal/nominal categorical predictors is suggested. The estimator is based on writing a conditional density as a sum of orthogonal projections on all possible subspaces of reduced dimensionality and then estimating each projection via a shrinkage procedure. The shrinkage procedure uses a universal thresholding and a dyadic-blockwise shrinkage for low and high…

概率密度与特征函数的自适应估计及预言不等式AI 解读

The theory of adaptive estimation and oracle inequalities for the case of Gaussian-shift–finite-interval experiments has made significant progress in recent years. In particular, sharp-minimax adaptive estimators and exact exponential-type oracle inequalities have been suggested for a vast set of functions including analytic and Sobolev with any positive index as well as for Efromovich–Pinsker and Stein blockwise-shrinkage estimators. Is it possible to obtain similar results…

回归场景中的条件密度估计AI 解读

Regression problems are traditionally analyzed via univariate characteristics like the regression function, scale function and marginal density of regression errors. These characteristics are useful and informative whenever the association between the predictor and the response is relatively simple. More detailed information about the association can be provided by the conditional density of the response given the predictor. For the first time in the literature, this article…

存在辅助协变量时的一元非参数回归AI 解读

This article addresses the problem of finding a relationship between the univariate predictor and the response when regression errors, created in part by known auxiliary covariates, are too large for a reliable regression estimation. A typical example is a controlled random design experiment with a large number of covariates, where the statistician is interested in the effect of a particular covariate and this effect is blurred by a large regression noise created by other…

回归误差密度的估计AI 解读

Estimation of the density of regression errors is a fundamental issue in regression analysis and it is typically explored via a parametric approach. This article uses a nonparametric approach with the mean integrated squared error (MISE) criterion. It solves a long-standing problem, formulated two decades ago by Mark Pinsker, about estimation of a nonparametric error density in a nonparametric regression setting with the accuracy of an oracle that knows the underlying…

偏倚抽样数据的密度估计AI 解读

The concept of biased data is well known and its practical applications range from social sciences and biology to economics and quality control. These observations arise when a sampling procedure chooses an observation with probability that depends on the value of the observation. This is an interesting sampling procedure because it favors some observations and neglects others. It is known that biasing does not change rates of nonparametric density estimation, but no results…

随机删失与序约束下的密度估计AI 解读

Do a random censorship and/or order restrictions (e.g., nonnegativity, monotonicity, convexity) affect estimation of a smooth density under mean integrated squared error (MISE)? Under mild assumptions, the known asymptotic results, which are concerned only with rates, answer “no.” This answer, especially for censored data, contradicts practical experience and statistical intuition. So what can be said about constants of MISE convergence? It is shown that asymptotically (a)…

摘要暂缺

准线性小波估计AI 解读

The main paradigm of the modern wavelet theory of spatial adaptation formulated by Donoho and Johnstone is that there is a divergence between the linear minimax adaptation theory and the heuristic guiding algorithm development that leads to the necessity of using strongly nonlinear adaptive thresholded methods. On the other hand, it is well known that linear adaptive estimates are the best whenever an estimated function is smooth. Is it possible to suggest a quasi-linear…

谱密度的数据驱动有效估计AI 解读

A nonparametric data-driven spectral density estimator is suggested for a class of processes with the exponentially decaying autocovariance function. This particular class is motivated by causal ARMA processes. The estimator is asymptotically efficient; that is, its mean integrated squared error converges with optimal minimax constant and rate as the sample size increases. The article also presents a Monte Carlo study of the estimator for the case of small sample sizes and…

数据驱动的有效谱密度估计AI 解读

A nonparametric data-driven spectral density estimator is suggested for a class of processes with the exponentially decaying autocovariance function. This particular class is motivated by causal ARMA processes. The estimator is asymptotically efficient; that is, its mean integrated squared error converges with optimal minimax constant and rate as the sample size increases. The article also presents a Monte Carlo study of the estimator for the case of small sample sizes and…

函数及其导数的同时精确估计AI 解读

A data-driven estimate is given that, over a Sobolev space, is simultaneously asymptotically sharp minimax for estimating both the function and its derivatives under integrated squared error loss. It is also shown that linear estimates cannot be simultaneously asymptotically sharp minimax over a given Sobolev space.

超光滑测量误差下的密度估计AI 解读

The problem is to estimate the probability density of a random variable contaminated by an independent measurement error. I explore one of the worst-case scenario when the characteristic function of this measurement error decreases exponentially and thus optimal estimators converge only with logarithmic rate. The particular example of such measurement error is any random variable contaminated by normal, Cauchy, or another stable random variable. For this setting and circular…

一般设定下独立同分布观测的非参数回归AI 解读

We consider the problem of sharp-optimal estimation of a response function $f(x)$ in a random design nonparametric regression under a general model where a pair of observations $(Y, X)$ has a joint density $p(y, x) = p(y|f(x)) \pi(x)$. We wish to estimate the response function with optimal minimax mean integrated squared error convergence as the sample size tends to $\infty$. Traditional regularity assumptions on the conditional density $p(y| \theta)$ assumed for parameter $\theta$…

二次泛函的最优自适应估计AI 解读

Minimax mean-squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes. In contrast to many nonparametric function estimation problems there are both regular and irregular cases. In the regular cases the minimax mean-squared error converges at a rate proportional to the inverse of the sample size, whereas in the irregular case much slower rates are the rule. We investigate the problem of adaptive estimation…

关于密度导数平方积分自适应估计的 Bickel–Ritov 猜想AI 解读

Bickel and Ritov suggested an optimal estimator for the integral of the square of the kth derivative of a density when the unknown density belongs to a Lipschitz class of a given order $\beta$. In this context optimality means that the estimate is asymptotically efficient, that is, it has the best constant and rate of risk convergence, whenever $\beta > 2k + 1/4$, and it is rate optimal otherwise. The suggested optimal estimator crucially depends on the value of $\beta$ which…

给定风险约束下的序贯非参数估计AI 解读

The problem is to estimate sequentially a nonparametric function known to belong to an $\alpha$-th-order Sobolev subspace $(\alpha > \frac{1}{2})$ with a minimax mean stopping time subject to an assigned maximum mean integrated squared error. For the case of a given $\alpha$ there exists a sharp estimator which has a minimal constant and a rate of minimax mean stopping time increasing as the assigned risk decreases. The situation changes drastically if $\alpha$ is unknown: a…