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生存分析中一类非参数检验的渐近性质AI 解读

Many of the popular nonparametric test statistics for censored survival data used in two-sample, $s$-sample trend and single continuous covariate situations are special cases of a general statistic, differing only in the choice of covariate-based label and weight function. Formulated in terms of counting processes and martingales this general statistic, standardized by the square root of its consistent variance estimator, is shown to be asymptotically normal under the null…

广类平稳、可能具有无限方差的线性过程经验分位数的 Bahadur 型表示AI 解读

Bahadur has obtained an asymptotic almost sure representation for empirical quantiles of independent and identically distributed random variables. In this paper we present an analogous result for a large class of stationary linear processes.

非线性计数过程回归模型的推断AI 解读

Martingale and counting process techniques are applied to the problem of inference for general conditional hazard functions. This problem was first studied by Beran, who introduced a class of estimators for the conditional cumulative hazard and survival functions in the special case of time-independent covariates. Here the covariate can be time dependent; the classical i.i.d. assumptions are relaxed by replacing them with certain asymptotic stability assumptions, and models…

扩散过程不完全观测渐近充分性的一个充分条件AI 解读

Consider an $m$-dimensional diffusion process $(X_t)$ with unknown drift and small known variance observed on a time interval $\lbrack 0, T\rbrack$. We derive here a general condition ensuring the asymptotic sufficiency, in the sense of Le Cam, of incomplete observations of $(X_t)_{0 \leq t \leq T}$ with respect to the complete observation of the diffusion as the variance goes to 0. We then construct estimators based on these partial observations which are consistent,…

随机过程参数估计中广义交叉验证与修正最大似然的比较AI 解读

Wahba compared the performance of generalized cross validation (GCV) and modified maximum likelihood (MML) procedures for choosing the smoothing parameter of a smoothing spline. This work makes a more careful study of the two procedures when the stochastic model motivating the modified maximum likelihood estimate is correct. In particular, it is shown that in the case of the linear smoothing spline with equally spaced observations, both estimates are asymptotically normal…

协方差函数错设时线性预测效率的界AI 解读

Suppose $z(\cdot)$ is a random process defined on a bounded set $R \subset \mathbb{R}^1$ with finite second moments. Consider the behavior of linear predictions based on $z(t_1), \ldots, z(t_n)$, where $t_1, t_2, \cdots$ is a dense sequence of points in $R$. Stein showed that if the second-order structure used to generate the predictions is incorrect but compatible with the correct second-order structure, the obtained predictions are uniformly asymptotically optimal as $n \rightarrow \infty$…

离子通道的更新理论性质AI 解读

The membranes of cells, such as those found in heart and nerve tissue, contain structures known as ion channels. The current produced by an ion channel in a cell membrane has been modelled as a finite state-space Markov process. Statistical analysis of ion channels has been largely limited to the case where a single channel is observed, or where the stimulus to the channel is so slight that only one channel at a time is emitting current. In many experiments, however, several…

Let $l(\theta) = n^{-1} \log p(x, \theta)$ be the log likelihood of an $n$-dimensional $X$ under a $p$-dimensional $\theta$. Let $\hat{\theta}_j$ be the mle under $H_j: \theta^1 = \theta^1_0, \ldots, \theta^j = \theta^j_0$ and $\hat{\theta}_0$ be the unrestricted mle. Define $T_j$ as $\lbrack 2n\{l(\hat{\theta}_{j - 1}) - l(\hat{\theta}_j)\}\rbrack^{1/2} \operatorname{sgn}(\hat{\theta}^j_{j - 1} - \theta^j_0).$ Let $T = (T_1, \ldots, T_p)$. Then under regularity conditions,…

关于函数典型变量、交替最小二乘法与 ACE 的讨论AI 解读

We discuss properties of some data-analytic methods which are intimately related to each other: alternating least squares (ALS), correspondence analysis and more recently Breiman and Friedman's ACE algorithm. The application of these methods to regression produces nonparametric estimators of nonlinear transformations, both of the response and the predictors. These procedures are among the most powerful tools for data analysis, but missing awareness of some artifacts could…

条件分位数的核估计与近邻估计AI 解读

Let $(X_1, Z_1), (X_2, Z_2), \ldots, (X_n, Z_n)$ be iid as $(X, Z), Z$ taking values in $R^1$, and for $0 < p < 1$, let $\xi_p(x)$ denote the conditional $p$-quantile of $Z$ given $X = x,$ i.e., $P(Z \leq \xi_p(x)\mid X = x) = p$. In this paper, kernel and nearest-neighbor estimators of $\xi_p(x)$ are proposed. In order to study the asymptotics of these estimates, Bahadur-type representations of the sample conditional quantiles are obtained. These representations are used to…

从柯尔莫哥洛夫逼近理论思想看密度估计AI 解读

The paper deals with upper and lower bounds for the quality of (probability) density estimation. Connections are established between these problems and the theory of approximation of functions. Particularly, it is demonstrated how some of Kolmogorov's concepts work.

格点分布的 Edgeworth 展开AI 解读

This paper investigates the use of Edgeworth expansions for approximating the distribution function of the normalized sum of $n$ independent and identically distributed lattice-valued random variables. We prove that the continuity-corrected Edgeworth series, using Sheppard-adjusted cumulants, is accurate to the same order in $n$ as the usual Edgeworth approximation for continuous random variables. Finally, as a partial justification of the Sheppard adjustments, it is shown…

正态分布方差的改进置信区间AI 解读

The usual confidence interval for the variance $\sigma^2$ of a normal distribution is a function of the sample variance alone. In this paper, we construct confidence intervals for $\sigma^2$ that also depend on the sample mean. These intervals have the same length as the shortest interval depending only on the sample variance and have uniformly higher probability of coverage. The coverage probabilities of these intervals and others are compared.

多元时间序列的典型偏自相关函数AI 解读

We propose a definition of the partial autocorrelation function $\beta(\cdot)$ for multivariate stationary time series suggested by the canonical analysis of the forward and backward innovations. Here $\beta(\cdot)$ satisfies $\beta(-n) = \beta(n)', n = 0, 1, \cdots,$ where $\beta(0)$ is nonnegative definite, $\{\beta(n), n = 1, 2, \cdots\}$ is a sequence of square matrices having singular values less than or equal to 1 and such that the order of $\beta(n + 1)$ is equal to…

爆炸性自回归中的弱自适应估计量AI 解读

Consider the model $X_i = \rho X_{i - 1} + \varepsilon_i, |\rho| > 1$, where $X_0, \varepsilon_1, \varepsilon_2, \cdots$ are independent random variables with $\varepsilon_1, \varepsilon_2, \cdots$ having common density $\psi$. This paper gives sufficient conditions under which the sequence of experiments induced by $\{X_0, X_1, \cdots, X_n\}$ has a weak limit in the sense of Le Cam. In general, the limiting experiment is translation invariant and neither LAN nor LAMN. The…

非参数与半参数模型中信息界的可达性AI 解读

We consider in this paper two widely studied examples of nonparametric and semiparametric models in which the standard information bounds are totally misleading. In fact, no estimators converge at the $n^{-\alpha}$ rate for any $\alpha > 0$, although the information is strictly positive "promising" that $n^{-1/2}$ is achievable. The examples are the estimation of $\int p^2$ and the slope in the model of Engle et al. A class of models in which the parameter of interest can be…

回归函数的估计AI 解读

In this paper, an entropy approach is proposed to establish rates of convergence for estimators of a regression function. General regression problems are considered, with linear regression, splines and isotonic regression as special cases. The estimation methods studied are least squares, least absolute deviations and penalized least squares. Common features of these methods and various regression problems are highlighted.

混合过程下非参数模型稳健估计量的渐近分布AI 解读

In this paper we obtain the asymptotic distribution of robust nonparametric autoregression estimators for dependent observations. Weights based on kernel and nearest-neighbor methods are considered.

相依数据核密度估计的数据驱动带宽选择AI 解读

The bandwidth selection problem in kernel density estimation is investigated in situations where the observed data are dependent. The classical leave-out technique is extended, and thereby a class of cross-validated bandwidths is defined. These bandwidths are shown to be asymptotically optimal under a strong mixing condition. The leave-one out, or ordinary, form of cross-validation remains asymptotically optimal under the dependence model considered. However, a simulation…

错误二阶结构下随机场线性预测的一致渐近最优性AI 解读

For a random field $z(t)$ defined for $t \in R \subseteq \mathbb{R}^d$ with specified second-order structure (mean function $m$ and covariance function $K$), optimal linear prediction based on a finite number of observations is a straightforward procedure. Suppose $(m_0, K_0)$ is the second-order structure used to produce the predictions when in fact $(m_1, K_1)$ is the correct second-order structure and $(m_0, K_0)$ and $(m_1, K_1)$ are "compatible" on $R$. For bounded $R$,…

估计混合密度与混合分布的傅里叶方法AI 解读

Let $X_1, X_2, \cdots$ be iid observations from a mixture density $f(x) = \int f(x \mid \theta)dG(\theta)$, where $f(x \mid \theta)$ is a known parametric family of density functions and $G$ is an unknown distribution function. This paper concerns estimating the mixing density $g = G'$ and the mixing distribution $G$. Fourier methods are used to derive kernel estimators, upper bounds for their rates of convergence and lower bounds for the optimal rate of convergence.…

Sobolev 类回归中的风险界AI 解读

For nonparametric regression estimation, when the unknown function belongs to a Sobolev smoothness class, sharp risk bounds for integrated mean square error have been found recently which improve on optimal rates of convergence results. The key to these has been the fact that under normality of the errors, the minimax linear estimator is asymptotically minimax in the class of all estimators. We extend this result to the nonnormal case, when the noise distribution is unknown.…

ACE 类方法的一致性AI 解读

The ACE (alternating conditional expectations) algorithm developed by Breiman and Friedman is an iterative method for finding optimal transformations of variables in multiple regression. Recently, several authors have extended ACE to discriminant analysis, time series and principal components. The central idea of ACE and of each of these extensions is that an optimal transformation $\phi^\ast$ minimizes a squared error-related functional over a Hilbert space, subject to…

对数样条模型的大样本推断AI 解读

Let $f$ be a continuous and positive unknown density on a known compact interval $\mathscr{Y}$. Let $F$ denote the distribution function of $f$ and let $Q = F^{-1}$ denote its quantile function. A finite-parameter exponential family model based on $B$-splines is constructed. Maximum-likelihood estimation of the parameters of the model based on a random sample of size $n$ from $f$ yields estimates $\hat{f, F}$ and $\hat{Q}$ of $f, F$ and $Q$, respectively. Under mild…