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观测 Fisher 信息的二阶最优性AI 解读

The realized error of an estimate is determined not only by the efficiency of the estimator, but also by chance. For example, suppose that we have observed a bivariate normal vector whose expectation is known to be on a circle. Then, intuitively, the longer that vector happens to be, the more accurately its angle is likely to be estimated. Yet this chance, though its information is contained in the data, cannot be accounted for by the variance of the estimate. One way to…

改进 Anscombe–Chow–Robbins 完全序贯抽样方案AI 解读

A new sequential sampling scheme is proposed in which, after an initial batch sample, sampling is continued in batches of data-dependent sizes (at most k such batches), and then one-at-a-time with a data-dependent stopping rule. This new scheme requires about the same sample size as the fully sequential Anscombe-Chow-Robbins (ACR) sampling scheme but substantially fewer sampling operations. The problem of constructing fixed-width confidence intervals for the mean of a normal…

自回归过程均值的有保证估计AI 解读

This paper considers the problem of sequential point estimation of the mean of a stable autoregressive process with unknown scale and autoregressive parameters. The construction of a sequential procedure makes use of special stopping rules and some modifications of least-squares estimates. The procedure enables estimating the mean with prescribed mean-square accuracy uniformly in nuisance parameters. The uniform asymptotic normality and the asymptotic minimaxity of the…

分布变化最优检测札记AI 解读

Suppose $X_1, X_2, \dots, X_{\nu - 1}$ are iid random variables with distribution $F_0$, and $X_{\nu}, X_{\nu + 1}, \dots$ are are iid with distributed $F_1$. The change point $\nu$ is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from $F_0$ to $F_1$ (detect the change), but to avoid false alarms. Pollak found a version of the Shiryayev-Roberts procedure to be asymptotically optimal for the problem of minimizing the average run…

弱相依过程的经验似然方法AI 解读

This paper studies the method of empirical likelihood in models with weakly dependent processes. In such cases, if the likelihood function is formulated as if the data process were independent, obviously empirical likelihood fails. We propose to use empirical likelihood of blocks of observations to solve this problem in a nonparametric manner. This method of "blockwise empirical likelihood" preserves the dependence of data, and the resulting likelihood ratios can be used to…

相关误差下非参数回归的大样本推断AI 解读

A central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum. We apply the result to kernel nonparametric fixed-design regression, giving a single central limit theorem which indicates how error spectral behavior at only zero frequency influences the asymptotic distribution and covers long-range, short-range and negative dependence. We show…

关于 Stahel–Donoho 估计量中 Tyler 的 MAD 修正AI 解读

In 1994 Tyler proposed a modification of the MAD in order to achieve the maximal possible finite sample breakdown point for the Stahel-Donoho estimator based upon the median and the modified MAD. A proof is not given, however, and as far as we know has not been published yet. Moreover there seems to be an error concerning the exact definition of the modified MAD which is needed to achieve this goal. For the sake of theoretical completeness and in order to correct this error…

关于奇异多元 Beta 分布与矩阵变换雅可比的 Uhlig 猜想证明AI 解读

Uhlig proposes two conjectures. The first concerns the Jacobian of the transformation $Y = B \times B'$ where B is the matrix $m \times m$ and m and X, Y belong to the class of positive semidefinite matrices of the order of $m \times m$ of rank $n < m, S_{m,n^{\cdot}}^+$. The second is concerned with the singular multivariate Beta distribution. This article seeks to prove the two conjectures. The latter result is then extended to the case of the singular multivariate F…

多元分布的 Zonoid 修剪AI 解读

A family of trimmed regions is introduced for a probability distribution in Euclidean d-space. The regions decrease with their parameter $\alpha$, from the closed convex hull of support (at $\alpha = 0$) to the expectation vector (at $\alpha = 1$). The family determines the underlying distribution uniquely. For every $\alpha$ the region is affine equivariant and continuous with respect to weak convergence of distributions. The behavior under mixture and dilation is studied.…

This paper presents two approaches for qualitative, quantitative and comparative concepts of skewness to be defined with respect to the spatial median for multivariate distributions. They extend the known quantile-based notions defined for real distributions. The main tool for such extensions consists of a family of central parts that provide suitable generalizations of the real interquantile intervals.

在对偶空间中用蒙特卡洛抽样近似经验半空间距离AI 解读

The Kolmogorov-Smirnov distance is an important tool for constructing confidence sets and tests in univariate problems. In multivariate settings, an analogous role is played by the halfspace distance, which has the merit of being invariant under linear transformations. However, the evaluation of the halfspace distance between two samples is a computationally very intensive combinatorial problem even in moderate dimensions, which severely restricts the use of the halfspace…

不规则设计下非线性小波回归的插值方法AI 解读

We introduce interpolation methods that enable nonlinear wavelet estimators to be employed with stochastic design, or nondyadic regular design, in problems of nonparametric regression. This approach allows relatively rapid computation, involving dyadic approximations to wavelet-after-interpolation techniques. New types of interpolation are described, enabling first-order variance reduction at the expense of second-order increases in bias. The effect of interpolation on…

CART 与最佳正交基：一种联系AI 解读

We study what we call "dyadic CART" --a method of nonparametric regression which constructs a recursive partition by optimizing a complexity penalized sum of squares, where the optimization is over all recursive partitions arising from midpoint splits. We show that the method is adaptive to unknown degrees of anisotropic smoothness. Specifically, consider the anisotropic smoothness classes of Nikoľskii, consisting of bivariate functions $f(x_1, x_2)$ whose finite difference…

REML估计中的Wald一致性与筛方法AI 解读

We prove that for all unconfounded balanced mixed models of the analysis of variance, estimates of variance components parameters that maximize the (restricted) Gaussian likelihood are consistent and asymptotically normal--and this is true whether normality is assumed or not. For a general (nonnormal) mixed model, we show estimates of the variance components parameters that maximize the (restricted) Gaussian likelihood over a sequence of approximating parameter spaces (i.e.,…

依赖指数半参数估计量收敛速率的说明AI 解读

Considerable recent attention has been devoted to semiparametric estimation of the dependence index, or the Hurst constant, using methods based on information in either frequency or time domains. Convergence rates of estimators in the frequency domain have been derived, and in the present paper we obtain them for estimators in the time domain. It is shown that the latter can have superior performance for moderate-range time series, but are inferior in the context of…

空间自回归中近单位根的渐近推断AI 解读

Asymptotic inference for estimators of $(\alpha_n, \beta_n)$ in the spatial autoregressive model $Z_{ij}(n) = \alpha_n Z_{i-1, j}(n) + \beta_n Z_{i, j-1}(n) - \alpha_n \beta_n Z_{i-1, j-1}(n) + \varepsilon_{ij}$ is obtained when $\alpha_n$ and $\beta_n$ are near unit roots. When $\alpha_n$ and $\beta_n$ are reparameterized by $\alpha_n = e^{c/n}$ and $\beta_n = e^{d/n}$, it is shown that if the "one-step Gauss-Newton estimator" of $\lambda_1 \alpha_n + \lambda_2 \beta_n$ is…

自动边界修正AI 解读

Many popular curve estimators based on smoothing have difficulties caused by boundary effects. These effects are visually disturbing in practice and can play a dominant role in theoretical analysis. Local polynomial regression smoothers are known to correct boundary effects automatically. Some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections. In both settings of density and regression estimation, we…

密度众数存在性的非参数检验AI 解读

Given a set of data drawn from an unknown density, it is frequently desirable to estimate the number and location of modes of the density. A test is proposed for the weight of evidence of individual observed modes. The test statistic used is a measure of the size of the mode, the absolute integrated difference between the estimated density and the same density with the mode in question excised at the level of the higher of its two surrounding antimodes. Samples are simulated…

删失数据的一类加权自助法研究AI 解读

Edgeworth expansions are derived for a class of weighted bootstrap methods for the Kaplan-Meier and Nelson-Aalen estimates using the methods contained in the monograph by Barbe and Bertail. Von Mises representations up to the third order are established for the weighted bootstrap versions of these estimators. It is shown that there exists weights which outperform Efron's bootstrap method in terms of coverage accuracy. Moreover, it is shown that this holds for a particular…

估计归一化常数比值的蒙特卡洛方法AI 解读

Recently, estimating ratios of normalizing constants has played an important role in Bayesian computations. Applications of estimating ratios of normalizing constants arise in many aspects of Bayesian statistical inference. In this article, we present an overview and discuss the current Monte Carlo methods for estimating ratios of normalizing constants. Then we propose a new ratio importance sampling method and establish its theoretical framework. We find that the ratio…

光滑函数积分的扰乱网格方差AI 解读

Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. One version, randomized $(t, m, s)$-nets, has the property that the integral estimates are unbiased and that the variance is $o(1/n)$, for any square integrable integrand. Stronger assumptions on the integrand allow one to find rates of convergence. This paper shows that for smooth integrands over…

平滑Cox回归AI 解读

Nonparametric regression was shown by Beran and McKeague and Utikal to provide a flexible method for analysis of censored failure times and more general counting processes models in the presence of covariates. We discuss application of kernel smoothing towards estimation in a generalized Cox regression model with baseline intensity dependent on a covariate. Under regularity conditions we show that estimates of the regression parameters are asymptotically normal at rate…

半参数似然比推断AI 解读

Likelihood ratio tests and related confidence intervals for a real parameter in the presence of an infinite dimensional nuisance parameter are considered. In all cases, the estimator of the real parameter has an asymptotic normal distribution. However, the estimator of the nuisance parameter may not be asymptotically Gaussian or may converge to the true parameter value at a slower rate than the square root of the sample size. Nevertheless the likelihood ratio statistic is…

风险回归中的局部似然与局部偏似然AI 解读

In survival analysis, the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model. This model usually assumes that the covariate has a log-linear effect on the hazard function. In this paper we consider the proportional hazards regression model with a nonparametric risk effect. We discuss estimation of the risk function and its derivatives in two cases: when the baseline hazard function is parametrized and…