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投影寻踪密度估计AI 解读

The projection pursuit methodology is applied to the multivariate density estimation problem. The resulting nonparametric procedure is often less biased than the kernel and near-neighbor methods. In addition, graphical information is produced that can be used to help gain geometric insight into the multivariate data distribution.

样条回归平滑参数选择的快速交叉验证方法AI 解读

The spline smoothing approach to nonparametric regression is considered, with particular reference to the problem of choosing how much to smooth. Both computational and statistical aspects of the method of cross-validation are discussed. An approximate cross-validation method is proposed that requires a small linear amount of computer time. A simulation study indicates that the method has very good statistical properties. Some mathematical justification and motivation are…

密度、回归曲线与众数的光滑最优核估计AI 解读

Several criteria concerning the choice of kernels for the nonparametric estimation of functions and their derivatives are discussed. A specific optimality criterion is described which applies to kernels of compact support and of different orders of smoothness. The solutions of the corresponding variational problems are explicitly given. Many kernels discussed previously are obtained as special cases.

非参数回归函数估计量的重对数律AI 解读

We study the estimation of a regression function by two classes of estimators, the Nadaraya-Watson Kernel type estimators and the orthogonal polynomial estimators. We obtain sharp pointwise rates of strong consistency by establishing laws of the iterated logarithm for the two classes of estimators. These results parallel those of Hall (1981) on density estimation and extend those of Noda (1976) on strong consistency of kernel regression estimators.

一类回归函数估计量的一致一致性AI 解读

We study a wide class of nonparametric regression function estimators including kernel estimators and robust smoothers. Under different assumptions on the kernel and the sequence of bandwidths, we obtain weak uniform consistency rates on a bounded interval. The uniform consistency is shown in a "stochastic design model" and in a "fixed design model".

含趋势与季节性的时间序列平滑先验状态空间建模AI 解读

A smoothness priors modeling of time series with trends and seasonalities is shown. An observed time series is decomposed into local polynomial trend, seasonal, globally stationary autoregressive and observation error components. Each component is characterized by an unknown variance–white noise perturbed difference equation constraint. The constraints or Bayesian smoothness priors are expressed in state space model form. Trading day factors are also incorporated in the…

一个非参数最大似然估计问题的简单解法AI 解读

Three approaches to a nonparametric maximum likelihood problem are considered. One, based on the method of "sieves", is shown to include the other two. The sieve considered is a double exponential convolution sieve. A closed form solution is given for certain values of the sieve parameter.

核型分位数估计量的相对亏损AI 解读

In this paper the asymptotic relative deficiency of the sample $q$-quantile with respect to kernel type estimators of the $q$-quantile is evaluated. The comparison is based on the mean square errors of the estimators. The result suggests a purely analytic measure of performance within the class of kernels. It is notable that a similar situation occurs when kernel estimators of a distribution function are studied.

回归函数核估计量的积分平方误差性质AI 解读

Weak laws of large numbers and central limit theorems are proved for integrated square error of kernel estimators of regression functions. The regressor is assumed to take values in $\mathbb{R}^p$, and the regressand, $X$, to be real valued. It is shown that in many cases, integrated square error is asymptotically normally distributed and independent of the $X$-sample. As an application, a test for the regression function (such as that proposed by Konakov) is seen to be…

生长曲线的非参数回归分析AI 解读

In recent years, nonparametric curve estimates have been extensively explored in theoretical work. There has, however, been a certain lack of convincing applications, in particular involving comparisons with parametric techniques. The present investigation deals with the analysis of human height growth, where longitudinal measurements were collected for a sample of boys and a sample of girls. Evidence is presented that kernel estimates of acceleration and velocity of height,…

一类贝叶斯非参数估计：第一部分，密度估计AI 解读

Given a positive, normalized kernel and a finite measure on an Euclidean space, we construct a random density by convoluting the kernel with the Dirichlet random probability indexed by the finite measure. The posterior distribution of the random density given a sample is classified. The Bayes estimator of the density function is given.

自回归依赖模型下核密度估计量的效率AI 解读

The problem of estimating the probability density function of a strictly stationary process is considered. To study the effect of a dependence structure on the efficiency of a kernel density estimator, the mean integrated squared error (MISE) of the Fourier integral estimator (FIE) is derived on the assumption that the observed data are generated by a first-order autoregressive process. Numerical results for the normal and Cauchy densities show that even moderate departures…

隐式统计模型推断的蒙特卡洛方法AI 解读

SUMMARY A prescribed statistical model is a parametric specification of the distribution of a random vector, whilst an implicit statistical model is one defined at a more fundamental level in terms of a generating stochastic mechanism. This paper develops methods of inference which can be used for implicit statistical models whose distribution theory is intractable. The kernel method of probability density estimation is advocated for estimating a log-likelihood from…

密度估计平滑参数的另一种交叉验证方法AI 解读

Cross-validation with Kullback-Leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation. A framework for this problem is constructed and used to derive an alternative method of cross-validation, based on integrated squared error, recently also proposed by Rudemo (1982). Hall (1983) has established the consistency and asymptotic optimality of the new method. For small and moderate sized samples, the performances…

样本谱分布泛函的极限行为AI 解读

The parameters of a stationary process can be viewed as functions of the spectral distribution function. This work concerns (estimators) parameters defined as integrals of $m (\geq 1)$-dimensional kernel functions with respect to the (sample) spectral distribution function. Conditions for asymptotic normality, almost sure convergence, and probability one bounds are derived for such estimators. The approach taken is based upon the reduction of an $m$-dimensional problem to…

核非参数回归中交叉验证的一致性AI 解读

For the nonparametric regression model $Y(t_i) = \theta(t_i) + \varepsilon(t_i)$ where $\theta$ is a smooth function to be estimated, $t_i$'s are nonrandom, $\varepsilon(t_i)$'s are i.i.d. errors, this paper studies the behavior of the kernel regression estimate $\hat{\theta}(t) = \big\lbrack \sum^n_{j=1}K \big(\frac{t_j - t}{\lambda}\big) Y(t_j) \big\rbrack / \big\lbrack\ sum^n_{j=1} K \big(\frac{t_j - t}{\lambda}\big) \big\rbrack$ when $\lambda$ is chosen by…

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混合比例的最小距离非参数估计AI 解读

SUMMARY Hall (1981a) develops a method of estimating mixture proportions using empirical distribution functions and a distance measure. A similar approach is suggested here based on density estimates. The method is applicable to general discrete data and multivariate data as well as to univariate continuous data and to ordered categorical data. For convenient quadratic distance measures the asymptotic theory is developed and remarks are made about practical implementation.…

交叉验证平滑样条的贝叶斯“置信区间”AI 解读

SUMMARY We consider the model Y(ti)=g(ti) + ∈i, i = 1, 2, …, n, where g(t), t ∈ [0, 1] is a smooth function and the {∈i} are independent N(0, σ2) errors with σ2 unknown. The cross-validated smoothing spline can be used to estimate g non-parametrically from observations on Y(ti), i= 1, 2, …, n, and the purpose of this paper is to study confidence intervals for this estimate. Properties of smoothing splines as Bayes estimates are used to derive confidence intervals based on…

随机删失数据风险函数变核估计的一则注记AI 解读

In a recent paper (Tanner and Wong, 1983b), a family of data-based nonparametric hazard estimators was introduced. Several of these estimators were studied in an extensive simulation experiment. The estimator which allows for variable bandwidth was found to have a superior performance. In this note, sufficient conditions for the variable kernel estimator to be strongly consistent are presented.

随机删失数据的风险函数核估计AI 解读

By convolution smoothing of the empirical hazards, a kernel estimate of the hazard function from censored data is obtained. Small and large sample expressions for the mean and the variance of the estimator are given. Conditions for asymptotic normality are investigated using the Hajek projection method.

核密度估计在L₁意义下弱收敛、强收敛与完全收敛的等价性AI 解读

Let $f$ be a density on $R^d$, and let $f_n$ be the kernel estimate of $f$, $f_n(x) = (nh^d)^{-1} \sum^n_{i=1} K((x - X_i)/h)$ where $h = h_n$ is a sequence of positive numbers, and $K$ is an absolutely integrable function with $\int K(x) dx = 1$. Let $J_n = \int |f_n(x) - f(x)| dx$. We show that when $\lim_nh = 0$ and $\lim_nnh^d = \infty$, then for every $\varepsilon > 0$ there exist constants $r, n_0 > 0$ such that $P(J_n \geq \varepsilon) \leq \exp(-rn), n \geq n_0$.…

广义条件下回归函数核估计的一致收敛AI 解读

In this article we prove uniform consistency of kernel estimators of a multivariate regression function under various assumptions on the distribution of the data. In addition to the usual assumptions that the data are iid and that the distribution of the regressors is absolutely continuous, we consider the cases that some regressors are discrete and the data are either stationary ϕ-mixing themselves or generated by a class of functions of one-sided infinite stationary…

M 型平滑样条的渐近理论AI 解读

Limit theorems giving rates of convergence of nonparametric regression estimates obtained from smoothing splines are proved. The main emphasis is on nonlinear, robust smoothing splines, but new results are obtained for the usual (linear) case. It is assumed that the knots become asymptotically uniform in a vague sense. Convergence of derivatives is also investigated. The main mathematical tools are a linearization of the robust smoothing spline, and an approximation of the…