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谱密度估计的筛方法AI 解读

We suggest a sieve for the estimation of the spectral density of a Gaussian stationary stochastic process. In contrast to the standard periodogram-based estimates this one aims at exploiting the full Gaussian nature of the process.

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关于球面样条的一个注记AI 解读

SUMMARY This paper presents two families of curves for interpolating and smoothing sequences of points on the surface of the unit sphere. One family is a natural analogue of the usual splines for curve-fitting in the plane; the other is slightly less optimal but more convenient from a computational viewpoint.

随机删失下风险率与密度函数的数据自适应核估计札记AI 解读

In a recent paper, Tanner (1983) proves pointwise consistency of a variable bandwidth kernel estimator for the hazard function. In the present note, a simplified proof of uniform consistency of a data-adaptive kernel estimator with certain additional advantages is given.

删失数据下风险率的直方图估计量AI 解读

A histogram type estimator of the hazard rate of lifetimes which are randomly right-censored is studied. This estimator is based on random spacings of the order statistics of uncensored observations and extends the idea of the histogram density estimation suggested by Van Ryzin (1973) to censored data. In this paper we establish pointwise large sample properties of the estimator including strong consistency, strong uniform consistency on a bounded interval, and two…

加性回归与其他非参数模型AI 解读

Let $(X, Y)$ be a pair of random variables such that $X = (X_1, \cdots, X_J)$ and let $f$ by a function that depends on the joint distribution of $(X, Y).$ A variety of parametric and nonparametric models for $f$ are discussed in relation to flexibility, dimensionality, and interpretability. It is then supposed that each $X_j \in \lbrack 0, 1\rbrack,$ that $Y$ is real valued with mean $\mu$ and finite variance, and that $f$ is the regression function of $Y$ on $X.$ Let $f^\ast,$…

频率多边形：理论与应用AI 解读

In this article I investigate the theoretical properties and applications of the frequency polygon, which is constructed by connecting with straight lines the mid-bin values of a histogram. For estimating an unknown probability density function using a random sample, the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error, achieving the same rate of convergence to zero of the integrated mean squared error as…

核分位数估计量的渐近正态性AI 解读

Multidimensional asymptotic normality of the kernel quantile estimator is established under fairly general conditions on the underlying distribution function and on the kernel. Sharpening these assumptions, one can utilize the proof to achieve also a bound for the rate of convergence which entails the comparison of the kernel estimator with the empirical quantile on the basis of their covering probabilities.

U 统计量的序贯非参数固定宽度置信区间AI 解读

A sequential fixed-width confidence interval for the mean of a $U$-statistic, having coverage probability approximately equal to preassigned $\alpha$, is presented. The main result, Theorem 2, shows that the sequential procedure is asymptotically efficient in the sense of Chow and Robbins (1965) and assumes only finiteness of the second moment of the kernel, the weakest possible condition. The paper follows naturally from Sproule (1974) and Sproule (1969), the primary…

过度平滑的非参数密度估计AI 解读

The optimal histogram for a sample of size n from a density defined on the entire real line requires at least (2n)1/3 bins, under mild smoothness conditions. Similar bounds exist for the frequency polygon and kernel estimators. Values near these bounds give nearly optimal results for a variety of smooth densities, providing good, quick density estimates.

平滑样条的诊断方法AI 解读

SUMMARY Diagnostic measures appropriate for use with smoothing splines are derived and their properties are investigated. The proposed measures focus on detection of observations which substantially influence the fit and provide additional information over that obtained from examination of residuals alone. A numerical example illustrates the technique.

非参数回归中若干带宽选择方法的渐近非等价性AI 解读

SUMMARY The bandwidth selection problem in nonparametric kernel regression is considered. Bandwidth selectors based on cross-validation and on Akaike's information criterion, atc, and his finite prediction error, FBE, are among those compared. It is seen that they are not necessarily asymptotically equivalent. Conditions are given under which the equivalence holds and modifications are suggested which make the selectors equivalent.

核回归估计的分布无关逐点一致性AI 解读

An estimate $\sum^n_{i=1} Y_iK((x - X_i)/h)/\sum^n_{j=1} K((x - X_j)/h)$, calculated from a sequence $(X_1, Y_1), \cdots, (X_n, Y_n)$ of independent pairs of random variables distributed as a pair $(X, Y)$, converges to the regression $E\{Y\mid X = x\}$ as $n$ tends to infinity in probability for almost all $(\mu) x \in R^d$, provided that $E|Y| < \infty, h \rightarrow 0$ and $nh^d \rightarrow \infty$ as $n \rightarrow \infty$. The result is true for all distributions $\mu$ o…

核密度估计的渐近最优带宽选择规则AI 解读

Kernel estimates of an unknown multivariate density are investigated, with mild restrictions being placed on the kernel. A window selection rule is considered, which can be interpreted in terms of cross-validation. Under the mild assumption that the unknown density and its one-dimensional marginals are bounded, the rule is shown to be asymptotically optimal. This strengthens recent results of Peter Hall.

一类非参数密度与回归估计量AI 解读

A class of maximum penalized likelihood estimators (MPLE) of the density function $f$ is constructed, through the use of a rather general roughness-penalty functional. This class contains all the density estimates in the literature that arise as solutions to MPLE problems with penalties on $f^{1/2}$. In addition, the flexibility of the penalty functional permits the construction of new spline estimates with improved performance at the peaks and valleys of the density curves.…

密度估计中的分布无关下界AI 解读

We consider the kernel estimate on the real line, $f_n(x) = (nh)^{-1} \sum^n_{i=1} K((X_i - x)/h),$ where $K$ is a bounded even density with compact support, and $X_1, \cdots, X_n$ are independent random variables with common density $f$. We treat the problem of placing a lower bound on the $L$1 error $J_n = E(\int |f_n - f|)$ which holds for all $f$. In particular, we show that there exist $A(K) \geq (9/125)^{1/5}$ depending only upon $K$, and $B^\ast(f) \geq 1$ depending…

自动核密度估计的一致性AI 解读

We consider the Parzen-Rosenblatt kernel density estimate on $\mathbb{R}^d$ with data-dependent smoothing factor. Sufficient conditions on the asymptotic behavior of the smoothing factor are given under which the estimate is pointwise consistent almost everywhere for all densities $f$ to be estimated. When the smoothing factor is a function only of the sample size $n$, it is shown that these conditions are also necessary, a generalization of results by Deheuvels. The…

非参数回归的带宽选择AI 解读

This paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression. We analyze a tapered Fourier series estimate and discuss the relationship of this estimate to a kernel estimate. We first consider a method based on an unbiased estimate of mean square error, and show that the bandwidth thus chosen is asymptotically optimal. Other methods are examined as well and are shown to be asymptotically equivalent. A small simulation shows,…

由二维截面估计三维肿瘤尺寸分布的交叉验证样条方法AI 解读

We study the problem of estimating the distribution of the three-dimensional radiuses of a collection of spheres, given measurements of the two-dimensional radiuses of a sample of planar cross sections. This problem arises in the estimation of the tumor size distribution of spherical microtumors induced in mouse livers following injection of a carcinogen. We first convert this problem to a form suitable for the application of cross-validated spline methods for the solution…

概率密度核估计量的一项最优性质AI 解读

SUMMARY Let f̂(x) be an asymptotically unbiased, nonparametric estimator of the density f(x) based on observations falling within the interval (x – ∈, x + ∈), where ∈ → 0 as the sample size diverges. We derive a lower bound to the variance of f(x), and prove that the bound is attained by a kernel estimator. This result is extended to estimates of f (r)(x), and to estimates of derivatives whose bias is improved by taking out the effects of unwanted terms.

稳健非参数函数拟合AI 解读

SUMMARY A robust non-parametric function fitting method is introduced. The estimate is motivated from the theory of M-estimation and of kernel estimation of regression functions. Consistency and asymptotic normality are shown. Bias and variance rates are the same as those previously obtained by Gasser and Müller (1979) for linear smoothers. The estimate satisfies a minimax property, i.e. it minimizes the maximal asymptotic variance as the error distributions vary over a…

最近邻回归函数估计的渐近正态性AI 解读

Let $(X, Y)$ be a random vector in the plane. We show that a smoothed N.N. estimate of the regression function $m(x) = \mathbb{E}(Y\mid X = x)$ is asymptotically normal under conditions much weaker than needed for the Nadaraya-Watson estimate. It also turns out that N.N. estimates are more efficient than kernel-type estimates if (in the mean) there are few observations in neighborhoods of $x$.

样条平滑：等价变带宽核方法AI 解读

The spline smoothing approach to nonparametric regression and curve estimation is considered. It is shown that, in a certain sense, spline smoothing corresponds approximately to smoothing by a kernel method with bandwidth depending on the local density of design points. Some exact calculations demonstrate that the approximation is extremely close in practice. Consideration of kernel smoothing methods demonstrates that the way in which the effective local bandwidth behaves in…

平滑先验与非线性回归AI 解读

Smoothness priors represent prior information that an unknown function does not change slope quickly and hence that the function describes a simple curve (e.g., Wahba 1978). In this article such priors for the multiple nonlinear regression model are developed in such a way that estimates and “standard errors” can be obtained as a natural and conceptually straightforward extension of linear multiple-regression estimation with the addition of dummy variables and dummy…