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We consider independent random elements $X_1, \ldots, X_n, n \in \mathbb{N}$, with values in a measurable space $(\mathscr{X}, \mathscr{B})$ so that $X_1, \ldots, X_{\lbrack n\theta\rbrack}$ have a common distribution $\nu_1$ and the remaining $X_{\lbrack n\theta\rbrack + 1}, \ldots, X_n$ have a common distribution $\nu_2 \neq \nu_1$, for some $\theta \in (0, 1)$. The change point $\theta$ as well as the distributions are unknown. A family of tests is introduced for the…

球面与欧氏空间中刚体运动的M估计AI 解读

This paper calculates the influence functions and asymptotic distributions of $M$-estimators of the rotation $A$ in a spherical regression model on the unit sphere in $p$ dimensions with isotropic errors. The problem arises in the reconstruction of the motion of a rigid body on the surface of the sphere. The comparable model for $p$-dimensional Euclidean space data is that $(\nu_1, \ldots, \nu_n)$ are independent with $\nu_i$ symmetrically distributed around $\gamma A \cdot u_i + b, u_i$…

U 分位数自助法的渐近精度AI 解读

The order of the Kolmogorov-Smirnov distance for the bootstrap of $U$-quantiles is considered. We observe that the order of the bootstrap of $U$-quantiles depends on the order of the local variance of the first term of the Hoeffding decomposition at the $U$-quantile. This order can be smaller than the order of the bootstrap of quantiles: $U$-quantiles can be smoother than quantiles.

Cox比例风险模型中抽样队列数据的分析方法AI 解读

Methods are provided for regression parameter and cumulative baseline hazard estimation in the Cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law. The key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes, leading to estimation methods parallel to those for full cohort data. These…

非参数标记依赖风险率模型中的核估计AI 解读

We introduce a new kernel hazard estimator in a nonparametric model where the stochastic hazard depends on the current value of time and on the current value of a time dependent covariate or marker. We establish the pointwise and global convergence of our estimator.

计数过程一般加性—乘性风险模型的半参数分析AI 解读

The additive-multiplicative hazard model specifies that the hazard function for the counting process associated with a multidimensional covariate process $Z = (W^T, X^T)^T$ takes the form of $\lambda(t\mid Z) = g\{\beta^T_0 W(t)\} + \lambda_0(t)h\{\gamma^T_0X(t)\}$, where $\theta_0 = (\beta^T_0, \gamma^T_0)^T$ is a vector of unknown regression parameters, $g$ and $h$ are known link functions and $\lambda_0$ is an unspecified "baseline hazard function." In this paper, we…

基于周期图的分形性质估计量AI 解读

We suggest an estimator, based on the periodogram, of the fractal index and fractal dimension of a continuous, stationary Gaussian process. We argue that the cosine part of the periodogram is more appropriate than the full periodogram for this application. The term "semiperiodogram" is used to describe the cosine component, and our estimator is based on simple linear regression of the logarithm of the semiperiodogram on the algorithm of frequency. Theoretical properties of…

不稳定长记忆过程的推断及其在分数单位根自回归中的应用AI 解读

An autoregressive time series is said to be unstable if all of its characteristic roots lie on or outside the unit circle, with at least one on the unit circle. This paper aims at developing asymptotic inferential schemes for an unstable autoregressive model generated by long-memory innovations. This setting allows both nonstationarity and long-memory behavior in the modeling of low-frequency phenomena. In developing these procedures, a novel weak convergence result for a…

干扰参数仅在备择假设下存在时似然比检验的可容许性AI 解读

This paper establishes the asymptotic admissibility of the likelihood ratio (LR) test for a general class of testing problems in which a nuisance parameter is present only under the alternative hypothesis. The paper also establishes the finite sample admissibility of the LR test for testing problems of this sort that arise in Gaussian linear regression models with known variance.

数据驱动平滑拟合优度检验的一致性与蒙特卡洛模拟AI 解读

The data driven method of selecting the number of components in Neyman's smooth test for uniformity, introduced by Ledwina, is extended. The resulting tests consist of a combination of Schwarz's Bayesian information criterion (BIC) procedure and smooth tests. The upper bound of the dimension of the exponential families in applying Schwarz's rule is allowed to grow with the number of observations to infinity. Simulation results show that the data driven version of Neyman's…

紧流形上若干概率分布的精确渐近分析AI 解读

Let $M$ be a compact, smooth, orientable manifold without boundary, and let $f: M \rightarrow \mathbb{R}$ be a smooth function. Let $dm$ be a volume form on $M$ with total volume 1, and denote by $X$ the corresponding random variable. Using a theorem of Kirwan, we obtain necessary conditions under which the method of stationary phase returns an exact evaluation of the characteristic function of $f(X)$. As an application to the Langevin distribution on the sphere $S^{d-1}$,…

统计估计的内蕴分析AI 解读

The parametric statistical models with suitable regularity conditions have a natural Riemannian manifold structure, given by the information metric. Since the parameters are merely labels for the probability measures, an inferential statement should be formulated through intrinsic objects, invariant under reparametrizations. In this context the estimators will be random objects valued on the manifold corresponding to the statistical model. In spite of these considerations,…

Wicksell问题中的保序估计与收敛速率AI 解读

It is shown that, in the nonparametric setting for the so-called Wicksell problem, the distribution function of the squared radii of the balls cannot be estimated at a rate faster than $n^{-1/2}\sqrt{\log n}$. We present an isotonic estimator of the distribution function which attains this rate and derive its asymptotic (normal) distribution. It is shown that the variance of this limiting distribution is exactly half the asymptotic variance of the naive plug-in estimator.

惩罚似然问题的变分解与平滑曲线估计AI 解读

Usual nonparametric regression estimators often show many little wiggles which do not appear to be necessary for a good description of the data. The new "Wp" smoother is a maximum penalized likelihood (MPL) estimate with a novel roughness penalty. It penalizes a relative change of curvature. This leads to disjoint classes of functions, each with given number, $n_w$, of inflection points. For a "Wp" estimate, $f"(x) = \pm (x - w_1)\cdots (x - w_{n_w}) \cdot \exp h_f(x)$,…

密度导数平方积分的自适应平方根样本量估计AI 解读

Based on a random sample of size $n$ from an unknown density $f$ on the real line, the nonparametric estimation of $\theta_k = \int\{f^{(k)}(x)\}^2 dx, k = 0, 1,\ldots$, is considered. These functionals are important in a number of contexts. The proposed estimates of $\theta_k$ is constructed in the frequency domain by using the sample characteristic function. It is known that the sample characteristic function at high frequency is dominated by sample variation and does not…

全局泛函的非参数估计与回归协变量解释力度量AI 解读

In a nonparametric regression setting with multiple random predictor variables, we give the asymptotic distributions of estimators of global integral functionals of the regression surface. We apply the results to the problem of obtaining reliable estimators for the nonparametric coefficient of determination. This coefficient, which is also called Pearson's correlation ratio, gives the fraction of the total variability of a response that can be explained by a given set of…

鞅的指数不等式及其在计数过程最大似然估计中的应用AI 解读

We obtain an exponential probability inequality for martingales and a uniform probability inequality for the process $\int g dN$, where $N$ is a counting process and where $g$ varies within a class of predictable functions $\mathscr{G}$. For the latter, we use techniques from empirical process theory. The uniform inequality is shown to hold under certain entropy conditions on $\mathscr{G}$. As an application, we consider rates of convergence for (nonparametric) maximum…

长程依赖的高斯半参数估计AI 解读

Assuming the model $f(\lambda) \sim G\lambda^{1-2H}$, as $\lambda \rightarrow 0 +$, for the spectral density of a covariance stationary process, we consider an estimate of $H \in (0, 1)$ which maximizes an approximate form of frequency domain Gaussian likelihood, where discrete averaging is carried out over a neighbourhood of zero frequency which degenerates slowly to zero as sample size tends to infinity. The estimate has several advantages. It is shown to be consistent…

偏离正态分布时位置参数标准置信区间的稳健性AI 解读

Let $X_i = \theta + \sigma Z_i$, where $Z_i$ are i.i.d. from a distribution $F$, and $-\infty < \theta < \infty$ and $\sigma > 0$ are unknown parameters. If $F$ is $N(0, 1)$, a standard confidence interval for the unknown mean $\theta$ is the $t$-interval $\bar{X} \pm t_{\alpha/2} s/\sqrt n$. The question of conservatism of this interval under nonnormality is considered by evaluating the infimum of its coverage probability when $F$ belongs to a suitably chosen class of…

任意维度置信集构造的新通用方法及其应用AI 解读

Let $\mathbf{X}$ have a star unimodal distribution $P_0$ on $\mathbb{R}^p$. We describe a general method for constructing a star-shaped set $S$ with the property $P_0(\mathbf{X} \in S) \geq 1 - \alpha$, where $0 < \alpha < 1$ is fixed. This is done by using the Camp-Meidell inequality on the Minkowski functional of an arbitrary star-shaped set $S$ and then minimizing Lebesgue measure in order to obtain size-efficient sets. Conditions are obtained under which this method…

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…

针对随机序增大备择的稳健序贯监测方案AI 解读

We sequentially observe independent observations $X_1, X_2,\ldots$ such that initially they have distribution $G_0$; at some unknown time $\nu$ they become stochastically larger, having distribution $G_1$. Neither $G_0$ nor $G_1$ is fully specified. We wish to detect that a change has taken place as soon as possible after its occurrence, subject to a constraint on the rate of false alarms. We derive a family of nonparametric sequential procedures based on ranks, with…

重尾随机变量归一化和的移动区块自助法渐近行为AI 解读

This paper studies the performance of the moving block bootstrap procedure for normalized sums of dependent random variables. Suppose that $X_1, X_2,\ldots$ are stationary $\rho$-mixing random variables with $\sum \rho (2^i) < \infty$. Let $T_n = (X_1 + \cdots + X_n - b_n)/a_n$, for some suitable constants $a_n$ and $b_n$, and let $T^\ast_{m,n}$ denote the moving block bootstrap version of $T_n$ based on a bootstrap sample of size $m$. Under certain regularity conditions, it…

迭代 Bootstrap 置信区间的渐近近似AI 解读

An iterated bootstrap confidence interval requires an additive correction to be made to the nominal coverage level of an uncorrected interval. Such correction is usually performed using a computationally intensive Monte Carlo simulation involving two nested levels of bootstrap sampling. Asymptotic expansions of the required correction and the iterated interval endpoints are used to provide two new computationally efficient methods for constructing an approximation to the…