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函数值空间过程的谱密度估计AI 解读

The spectral density function describes the second-order properties of a stationary stochastic process on Rd. This paper considers the nonparametric estimation of the spectral density of a continuous-time stochastic process taking values in a separable Hilbert space. Our estimator is based on kernel smoothing and can be applied to a wide variety of spatial sampling schemes including those in which data are observed at irregular spatial locations. Thus, it finds immediate…

具有多重不连续点的函数型过程AI 解读

Abstract We consider the problem of estimating multiple change points for a functional data process. There are numerous examples in science and finance in which the process of interest may be subject to some sudden changes in the mean. The process data that are not in a close vicinity of any change point can be analysed by the usual nonparametric smoothing methods. However, the data close to change points and contain the most pertinent information of structural breaks need…

Hurst 函数估计AI 解读

This paper considers a wide range of issues concerning the estimation of the Hurst function of a multifractional Brownian motion when the process is observed on a regular grid. A theoretical lower bound for the minimax risk of this inference problem is established for a wide class of smooth Hurst functions. We also propose a new nonparametric estimator and show that it is rate optimal. Implementation issues of the estimator including how to overcome the presence of a…

局部内蕴平稳性及其统计推断AI 解读

Dense spatial data are commonplace nowadays, and they provide the impetus for addressing nonstationarity in a general way. This paper extends the notion of intrinsic random function by allowing the stationary component of the covariance to vary with spatial location. A nonparametric estimation procedure based on gridded data is introduced for the case where the covariance function is regularly varying at any location. An asymptotic theory is developed for the procedure on a…

函数型与纵向数据中非参数回归和主成分分析的一致收敛速率AI 解读

We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of…

函数型与高维数据有效降维空间的维数判定AI 解读

In this paper, we consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is called the effective dimension reduction (EDR) space. To determine the dimensionality of the EDR space, we focus on the leading principal component scores of the predictor, and propose two sequential χ2 testing procedures under the…

逆回归降维问题的再生核希尔伯特空间表述AI 解读

Suppose that Y is a scalar and X is a second-order stochastic process, where Y and X are conditionally independent given the random variables ξ1, …, ξp which belong to the closed span LX2 of X. This paper investigates a unified framework for the inverse regression dimension-reduction problem. It is found that the identification of LX2 with the reproducing kernel Hilbert space of X provides a platform for a seamless extension from the finite- to infinite-dimensional settings.…

最近邻逆回归AI 解读

Sliced inverse regression (SIR), formally introduced by Li, is a very general procedure for performing dimension reduction in nonparametric regression. This paper considers a version of SIR in which the “slices” are determined by nearest neighbors and the response variable takes value possibly in a multidimensional space. It is shown, under general conditions, that the “effective dimension reduction space” can be estimated with rate $n^{-1/2}$ where $n$ is the sample size.

长记忆移动平均过程经验过程的渐近展开AI 解读

Let $X_n = \Sigma_{i=1}^{\infty} a_i \varepsilon_{n-i}$, where the $\varepsilon_i$ are iid with mean 0 finite fourth moment and the $a_i$ are regularly varying with index $-\beta$ where $\beta \epsilon (1/2, 1)$ so that ${X_n}$ has long-range dependence. This covers an important class of the fractional ARIMA process. For $r \geq 0$, let $Y_{N, r} = \sum_{n=1}^N \sum_{1\leq j_1 < \dots < j_r} \Pi_{s=1}^r a_{j_s}, Y_{N, 0} = N, \sigma_{N, r}^2 = \Var(Y_{N, r})$ and $F^{(r)} =$…

弱依赖平稳序列的极值指数估计AI 解读

Under stationarity and weak dependence, the statistical significance and the estimation of the extremal index are considered. It is shown that the distribution of the sample maximum can be uniformly approximated given the extremal index and the marginal distribution as the sample size increases. An adaptive procedure is proposed for estimating the extremal index. The procedure is shown to be asymptotically optimal in a class of estimators.

切片逆回归的渐近理论AI 解读

Sliced inverse regression [Li (1989), (1991) and Duan and Li (1991)] is a nonparametric method for achieving dimension reduction in regression problems. It is widely applicable, extremely easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. If $Y$ is the response and $X \in \mathbf{R}^p$ is the predictor, in order to implement sliced inverse regression, one requires an estimate of $\Lambda = E\{\operatorname{cov}(X\mid Y)\} = \operatorname{cov}(X) - \operatorname{cov}\{E(X\mid Y)\}$…

依赖数据下的尾指数估计AI 解读

Let $X_1, X_2,\ldots$ be possibly dependent random variables having the same marginal distribution. Consider the situation where $\bar{F}(x) := P\lbrack X_1 > x\rbrack$ is regularly varying at $\infty$ with an unknown index $- \alpha < 0$ which is to be estimated. In the i.i.d. setting, it is well known that Hill's estimator is consistent for $\alpha^{-1}$, and is asymptotically normally distributed. It is the purpose of this paper to demonstrate that such properties of…