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更接近最优值的回归平滑参数AI 解读

Abstract It is well known that data-driven regression smoothing parameters ħ based on cross-validation and related methods exhibit a slow rate of convergence to their optimum. In an earlier article we showed that this rate can be as slow as n –1/10; that is, for a bandwidth ħ 0 optimizing the averaged squared error, n 1/10 (ħ — ħ 0)/ħ 0 tends to an asymptotic normal distribution. In this article we consider mean averaged squared error optimal bandwidths h 0. This (nonrandom)…

平均导数估计的带宽选择AI 解读

Abstract The average derivative is the expected value of the derivative of a regression function. Kernel methods have been proposed as a means of estimating this quantity. The problem of bandwidth selection for these kernel estimators is addressed here. Asymptotic representations are found for the variance and squared bias. These are compared with each other to find an insightful representation for a bandwidth optimizing terms of lower order than n –1. It is interesting…

接近最优值的回归平滑参数AI 解读

It is well known that data-driven regression smoothing parameters h based on cross-validation and related methods exhibit a slow rate of convergence to their optimum. In an earlier article we showed that this rate can be as slow as n-1/10; that is, for a bandwidth h0 optimizing the averaged squared error, n1/10 (h - h0)/h0 tends to an asymptotic normal distribution. In this article we consider mean averaged squared error optimal bandwidths h0. This (nonrandom) smoothing…

平均导数估计的带宽选择AI 解读

The average derivative is the expected value of the derivative of a regression function. Kernel methods have been proposed as a means of estimating this quantity. The problem of bandwidth selection for these kernel estimators is addressed here. Asymptotic representations are found for the variance and squared bias. These are compared with each other to find an insightful representation for a bandwidth optimizing terms of lower order than n-1. It is interesting that, for…

Reinsch 样条与 Speckman 样条的比较AI 解读

In the context of nonparametric regression, spline smoothers can be used both for the estimation of the curve itself and for the estimation of the error variance. We compare two approaches that have been suggested: the Reinsch spline, i.e. roughness penalty, and the Speckman spline, minimax. The comparison is made on the basis of the summed mean squared error for curve estimation and on the basis of a closely related criterion for variance estimation. We show in both…

经验泛函与高效平滑参数选择AI 解读

SUMMARY A striking feature of curve estimation is that the smoothing parameter ĥ 0, which minimizes the squared error of a kernel or smoothing spline estimator, is very difficult to estimate. This is manifest both in slow rates of convergence and in high variability of standard methods such as cross-validation. We quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of ĥ 0 that attain the bounds. The…

对条件均值误设稳健的异方差检验AI 解读

This paper proposes a new test statistic to deter the presence of heteroskedasticity. The proposed test does not require a parametric specification of the mean regression function in the first stage regression. The regression function is estimated nonparametrically by the kernel estimation method. The nonparametric residual is estimated and used as a proxy for the random disturbance term. This nonparametric residual is robust to regression function misspecification.…

残差相关时核回归的带宽选择AI 解读

Standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated. In this paper we propose a modified version of an existing plug-in bandwidth selector. The method is generalized to stationary error variables by estimating a functional of the residual covariance function. The proposed bandwidth selector shows good properties in asymptotic theory and in simulations without…

核密度估计的自动带宽选择器AI 解读

To select a proper bandwidth is a critical step in kernel density estimation. It is well known that the bandwidth selected by cross-validation has a large variability. This difficulty limits the applicability of cross-validation. To reduce the variability, we suggested modifying the sample characteristic function beyond some cut-off frequency in estimating the bias term of the mean integrated squared error. It is proposed to select the cut-off frequency by a generalization…

左截断与右删失数据的风险率估计量比较AI 解读

Left truncation and right censoring arise frequently in practice for life data. This paper is concerned with the estimation of the hazard rate function for such data. Two types of nonparametric estimators based on kernel smoothing methods are considered. The first one is obtained by convolving a kernel with a cumulative hazard estimator. The second one is in the form of a ratio of two statistics. Local properties including consistency, asymptotic normality and mean squared…

具有 ARMA 误差的非参数样条回归AI 解读

We estimate by spline nonparametric regression an unknown function observed with autocorrelated errors when the errors are modelled by an autoregressive moving average model. Unknown parameters are estimated by either maximum likelihood, cross-validation or generalized cross-validation. By expressing the problem in state space form we obtain O(n) algorithms to estimate the function and its derivatives and evaluate the marginal likelihood and cross-validation functions. The…

Reinsch 与 Speckman 样条的比较AI 解读

In the context of nonparametric regression, spline smoothers can be used both for the estimation of the curve itself and for the estimation of the error variance. We compare two approaches that have been suggested: the Reinsch spline, i.e. roughness penalty, and the Speckman spline, minimax. The comparison is made on the basis of the summed mean squared error for curve estimation and on the basis of a closely related criterion for variance estimation. We show in both…

一种简单的根号 n 带宽选择器AI 解读

The asymptotically best bandwidth selectors for a kernel density estimator currently require the use of either unappealing higher order kernel pilot estimators or related Fourier transform methods. The point of this paper is to present a methodology which allows the fastest possible rate of convergence with the use of only nonnegative kernel estimators at all stages of the selection process. The essential idea is derived through careful study of factorizations of the pilot…

相关误差下两种带宽选择方法的比较AI 解读

For nonparametric regression, in the case of dependent observations, cross-validation is known to be severely affected by dependence. This effect is precisely quantified through a limiting distribution for the cross-validated bandwidth. The performance of two methods, the "leave-$(2l + 1)$-out" version of cross-validation and partitioned cross-validation, which adjust for the effect of dependence on bandwidth selection is investigated. The bandwidths produced by these two…

核密度估计的带宽选择AI 解读

The problem of automatic bandwidth selection for a kernel density estimator is considered. It is well recognized that the bandwidth estimate selected by the least squares cross-validation is subject to large sample variation. This difficulty limits the application of the cross-validation estimate. Based on characteristic functions, an important expression for the cross-validation bandwidth estimate is obtained. The expression clearly points out the source of variation. To…

连续协变量非参数回归中的变量选择AI 解读

In a nonparametric regression setup where the covariates are continuous, the problem of estimating the number of covariates will be discussed in this paper. The kernel method is used to estimate the regression function and the selection criterion is based on minimizing the cross-validation estimate of the mean squared prediction error. We consider choosing both the bandwidth and the number of covariates based on the data. Unlike the case of linear regression, it turns out…

双重灵活对数样条响应模型的渐近理论AI 解读

Consider a $\mathscr{Y}$-valued response variable having a density function $f(\cdot\mid x)$ that depends on an $\mathscr{X}$-valued input variable $x.$ It is assumed that $\mathscr{X}$ and $\mathscr{Y}$ are compact intervals and that $f(\cdot\mid\cdot)$ is continuous and positive on $\mathscr{X} \times \mathscr{Y}.$ Let $F(\cdot\mid x)$ denote the distribution function of $f(\cdot\mid x)$ and let $Q(\cdot\mid x)$ denote its quantile function. A finite-parameter exponential…

随机删失下核密度估计Lp距离的中心极限定理AI 解读

A sequence of independent nonnegative random variables with common distribution function $F$ is censored on the right by another sequence of independent identically distributed random variables. These two sequences are also assumed to be independent. We estimate the density function $f$ of $F$ by a sequence of kernel estimators $f_n(t) = (\int^\infty_{-\infty}K((t - x)/h(n))d\hat{F}_n(x))/h(n),$ where $h(n)$ is a sequence of numbers, $K$ is kernel density function and $\hat{F}_n$…

交互样条模型及其收敛速率AI 解读

We consider interaction splines which model a multivariate regression function $f$ as a constant plus the sum of functions of one variable (main effects), plus the sum of functions of two variables (two-factor interactions), and so on. The estimation of $f$ by the penalized least squares method and the asymptotic properties of the models are studied in this article. It is shown that, under some regularity conditions on the data points, the expected squared error averaged…

样条平滑参数的交叉验证与最大似然估计性能AI 解读

An important aspect of nonparametric regression by spline smoothing is the estimation of the smoothing parameter. In this article we report on an extensive simulation study that investigates the finite-sample performance of generalized cross-validation, cross-validation, and marginal likelihood estimators of the smoothing parameter in splines of orders 2 and 3. The performance criterion for both the estimate of the function and its first derivative is measured by the square…

确定性函数的贝叶斯预测及计算机试验设计AI 解读

This article is concerned with prediction of a function y(t) over a (multidimensional) domain T, given the function values at a set of “sites” {t (1), t (2), …, t (n)} in T, and with the design, that is, with the selection of those sites. The motivating application is the design and analysis of computer experiments, where t determines the input to a computer model of a physical or behavioral system, and y(t) is a response that is part of the output or is calculated from it.…

利用多元自适应回归样条进行非线性时间序列建模AI 解读

Multivariate Adaptive Regression Splines (MARS) is a new methodology, due to Friedman, for nonlinear regression modeling. MARS can be conceptualized as a generalization of recursive partitioning that uses spline fitting in lieu of other simple fitting functions. Given a set of predictor variables, MARS fits a model in the form of an expansion in product spline basis functions of predictors chosen during a forward and backward recursive partitioning strategy. MARS produces…

样条平滑参数的交叉验证与最大似然估计表现AI 解读

An important aspect of nonparametric regression by spline smoothing is the estimation of the smoothing parameter. In this article we report on an extensive simulation study that investigates the finite-sample performance of generalized cross-validation, cross-validation, and marginal likelihood estimators of the smoothing parameter in splines of orders 2 and 3. The performance criterion for both the estimate of the function and its first derivative is measured by the square…

高效的偏差校正非参数谱估计AI 解读

We consider estimation of a spectral density at a particular frequency. A linear model is developed based on the local second order behaviour of kernel spectral estimates and a second stage estimate is derived from this model. The estimate is shown to be asymptotically more efficient than the kernel estimate with the optimal, but unknown, bandwidth whenever the smoothing parameter is chosen so that it oversmooths the kernel estimate. A small-sample simulation study shows…