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修正最大对称 2s 间距的极限行为及应用AI 解读

This paper presents a solution to an open problem in astrophysics, namely that of estimating nonparametrically the strength of the pulsed signal in a series of high-energy photon arrival times. The newly proposed estimator, based on a modified maximal symmetric $2s$-spacing, is shown to be strongly consistent and asymptotically normally distributed, and a Monte Carlo study shows that its small and moderate sample behavior is very satisfactory. Additionally, new results…

区块自助法的理论比较AI 解读

In this paper, we compare the asymptotic behavior of some common block bootstrap methods based on nonrandom as well as random block lengths. It is shown that, asymptotically, bootstrap estimators derived using any of the methods considered in the paper have the same amount of bias to the first order. However, the variances of these bootstrap estimators may be different even in the first order. Expansions for the bias, the variance and the mean-squared error of different…

时间序列的非参数模型检验AI 解读

This paper studies a class of tests useful for testing the goodness-of-fit of an autoregressive model. These tests are based on a class of empirical processes marked by certain residuals. The paper first gives their large sample behavior under null hypotheses. Then a martingale transformation of the underlying process is given that makes tests based on it asymptotically distribution free. Consistency of these tests is also discussed briefly.

密度估计中 Dirichlet 混合的后验一致性AI 解读

A Dirichlet mixture of normal densities is a useful choice for a prior distribution on densities in the problem of Bayesian density estimation. In recent years, efficient Markov chain Monte Carlo method for the computation of the posterior distribution has been developed. The method has been applied to data arising from different fields of interest. The important issue of consistency was however left open. In this paper, we settle this issue in affirmative.

随机逼近版 EM 算法的收敛性AI 解读

The expectation-maximization (EM) algorithm is a powerful computational technique for locating maxima of functions. It is widely used in statistics for maximum likelihood or maximum a posteriori estimation in incomplete data models. In certain situations, however, this method is not applicable because the expectation step cannot be performed in closed form. To deal with these problems, a novel method is introduced, the stochastic approximation EM (SAEM), which replaces the…

删失回归数据的降维AI 解读

Without parametric assumptions, high-dimensional regression analysis is already complex. This is made even harder when data are subject to censoring. In this article, we seek ways of reducing the dimensionality of the regressor before applying nonparametric smoothing techniques. If the censoring time is independent of the lifetime, then the method of sliced inverse regression can be applied directly. Otherwise, modification is needed to adjust for the censoring bias. A key…

Sobolev 类函数的逐点与上确界范数精确自适应估计AI 解读

The problem of nonparametric function estimation in the Gaussian white noise model is considered. It is assumed that the unknown function belongs to one of the Sobolev classes, with an unknown regularity parameter. Asymptotically exact adaptive estimators of functions are proposed on the scale of Sobolev classes, with respect to pointwise and sup-norm risks. It is shown that, unlike the case of $L_2$-risk, a loss of efficiency under adaptation is inevitable here. Bounds on…

Wicksell 问题中的渐近有效估计AI 解读

We consider the classical Wicksell problem of estimating an unknown distribution function $G$of the radii of balls, based on their observed crosssections. It is assumed that the underlying distribution function$G$ belongs to a Hölder class of smoothness $\gamma >1/2$. We prove that, for a suitable choice of the smoothing parameters, kernel-type estimators are asymptotically efficient for a large class of symmetric bowl-shaped loss functions.

非参数反卷积的保序逆估计量AI 解读

A new nonparametric estimation procedure is introduced for the distribution function in a class of deconvolution problems, where the convolution density has one discontinuity. The estimator is shown to be consistent and its cube root asymptotic distribution theory is established. Known results on the minimax risk for the estimation problem indicate the estimator to be efficient.

经验鞍点与自助法尾部近似的小相对误差概率界AI 解读

To obtain test probabilities based on empirical approximations to the distribution of a Studentized function of a mean, we need the approximations to be accurate with sufficiently high probability. In particular, when these test probabilities are small it is best to consider relative errors. Here we show that in the case of univariate standardized means and in the general case of tests based on smooth functions of means, the empirical approximations have asymptotically small…

多个回归函数的非参数比较：精确与渐近理论AI 解读

A new test is proposed for the comparison of two regression curves $f$ and $g$. We prove an asymptotic normal law under fixed alternatives which can be applied for power calculations, for constructing confidence regions and for testing precise hypotheses of a weighted $L_2$ distance between $f$ and $g$ . In particular, the problem of nonequal sample sizes is treated, which is related to a peculiar formula of the area between two step functions. These results are extended in…

多元模型中单侧备择的方向性检验AI 解读

Consider one-sided testing problems for a multivariate exponential family model. Through conditioning or other considerations, the problem oftentimes reduces to testing a null hypothesis that the natural parameter is a zero vector against the alternative that the natural parameter lies in a closed convex cone $\mathscr{C}$. The problems include testing homogeneity of parameters, testing independence in contingency tables, testing stochastic ordering of distributions and many…

We study integration and reconstruction of Gaussian random functions with inhomogeneous local smoothness. A single realization may only be observed at a finite sampling design and the correct local smoothness is unknown. We construct adaptive two-stage designs that lead to asymptotically optimal methods. We show that every nonadaptive design is less efficient.

利用时空数据估计随机过程的积分AI 解读

Consider a space–time stochastic process $Z_ t(x)=S(x) + \xi_t(x)$ where $S(x)$ is a signal process defined on $\mathbb{R}^d$ and $\xi_t(x)$ represents measurement errors at time $t$. For a known measurable function $v(x)$ on $\mathbb{R}^d$ and a fixed cube $ D\subset \mathbb{R}^d$, this paper proposes a linear estimator for the stochastic integral $\int_D v(x)S(x)dx$ based on space–time observations $\{Z_t(x_i); i = 1,\ldots,n; t=1,\ldots,T\}$. Under mild conditions, the…

不连续回归曲面的拟合AI 解读

We suggest a three-stage procedure to recover discontinuous regression surfaces when noisy data are present. In the first stage, jump candidate points are detected using a jump detection criterion. A local principal component line is then fitted through these points in a neighborhood of a design point. This line provides a first-order approximation to the true jump location curve in that neighborhood. In the third stage, observations on the same side of the line as the given…

规则网格数据下光滑曲线近似的全局表现AI 解读

Approximating boundaries using data recorded on a regular grid induces discrete rounding errors in both vertical and horizontal directions. In cases where grid points exhibit at least some degree of randomness, an extensive theory has been developed for local-polynomial boundary estimators. It is inapplicable to regular grids, however. In this paper we impose strict regularity of the grid and describe the performance of local linear estimators in this context. Unlike the…

有限半参数混合模型中广义Pólya urn方案的指数后验一致性AI 解读

Advances in Markov chain Monte Carlo MCMC methods now make it computationally feasible and relatively straightforward to apply the Dirichlet process prior in a wide range of Bayesian nonparametric problems. The feasibility of these methods rests heavily on the fact that the MCMC approach avoids direct sampling of the Dirichlet process and is instead based on sampling the finite-dimensional posterior which is obtained from marginalizing out the process. In application, it is…

Gibbs 采样器的信息界AI 解读

If we wish to estimate efficiently the expectation of an arbitrary function on the basis of the output of a Gibbs sampler, which is better: deterministic or random sweep? In each case we calculate the asymptotic variance of the empirical estimator, the average of the function over the output, and determine the minimal asymptotic variance for estimators that use no information about the underlying distribution. The empirical estimator has noticeably smaller variance for…

多维均值泛函的贝叶斯自助法可信集AI 解读

This paper shows that the Bayesian bootstrap (BB) distribution of a multidimensional mean functional based on i.i.d. observations has a strongly unimodal Lebesgue density provided the convex hull of the data has a nonempty interior. This result is then used to construct the finite sample BB credible sets. The influence of an outlier on these credible sets is studied in detail and a comparison is made with the empirical likelihood ratio confidence sets in this context.

最大似然先验AI 解读

Consider an estimate $\theta^*$ of a parameter $\theta$ based on repeated observations from a family of densities $f_\theta$ evaluated by the Kullback–Leibler loss function $K(\theta, \theta^*) = \int \log(f_\theta/f_{\theta^*})f_\theta$. The maximum likelihood prior density, if it exists, is the density for which the corresponding Bayes estimate is asymptotically negligibly different from the maximum likelihood estimate. The Bayes estimate corresponding to the maximum…

多元分布离散度矩阵的极小极大偏差稳健估计AI 解读

Maronna defines affine equivariant $M$-estimators for multivariate location and scatter. They are particularly suited for estimating the pseudo-covariance or scatter matrix of an elliptical population. By defining the bias of a dispersion matrix properly, we consider the maximum bias of an $M$-estimator over an $\varepsilon$ -neighborhood of the underlying elliptical distribution (location known). We find that Tyler’s estimator minimizes the maximum bias.

轮廓估计、集中函数与序约束下的最大似然密度估计AI 解读

Based on empirical Lévy-type concentration functions, a new graphical representation of the ML-density estimator under order restrictions is given. This representation generalizes the well-known representation of the Grenander estimator of a monotone density as the slope of the least concave majorant of the empirical distribution function to higher dimensions and arbitrary order restrictions. From the given representation it follows that a density estimator called…

估计量与置信集的自适应调节AI 解读

An unknown signal plus white noise is observed at $n$ discrete time points. Within a large convex class of linear estimators of $\xi$, we choose the estimator $\hat{\xi}$ that minimizes estimated quadratic risk. By construction, $\hat{\xi}$ is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coefficients of the transformed data. If the class of candidate estimators satisfies…

重尾过程的样本自相关及其在 ARCH 模型中的应用AI 解读

We study the sample ACVF and ACF of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying. This includes linear and bilinear processes with regularly varying noise and ARCH processes, their squares and absolute values. We show that the distributional limits of the sample ACF can be random, provided that the variance of the marginal distribution is infinite and the process is nonlinear. This is in…