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线性回归模型中百分位数的置信带AI 解读

Simultaneous confidence intervals for percentiles of the normal regression model similar to those given by Steinhorst and Bowden (1971) are considered. Kanofsky's (1968) confidence band for a single normal distribution is modified and extended to the regression model. The confidence bands that are simultaneous in all percentiles provide corresponding confidence bands for the cumulative conditional normal distribution functions. The various procedures are compared with…

离散判别分析中核密度估计的另一种平滑选择AI 解读

The kernel method of estimating the cell probabilities of a multivariate categorical distribution, due to Aitchison & Aitken (1976), depends crucially on an unknown smoothing parameter λ. A method of estimating A is introduced which is explicitly connected to multivariate discrimination. The method, based on maximization of the leaving-one-out estimator of the nonerror rate, is shown to be Bayes risk strongly consistent. An example is given to illustrate the application.

删失数据下核密度估计量的若干渐近性质AI 解读

The kernel estimator is a widely used tool for the estimation of a density function. In this paper its adaptation to censored data using the Kaplan-Meier estimator is considered. Asymptotic properties of four estimators, arising naturally as a result of considering various types of bandwidths, are investigated. In particular we show that (i) both proposed estimators stemming from the nearest neighbor estimator have censoring-free variances and (ii) one of them is pointwise…

广义加性模型的降维原理AI 解读

Let $(X, Y)$ be a pair of random variables such that $X = (X_1,\cdots, X_J)$ ranges over $C = \lbrack 0, 1\rbrack^J$. The conditional distribution of $Y$ given $X = x$ is assumed to belong to a suitable exponential family having parameter $\eta \in \mathbb{R}$. Let $\eta = f(x)$ denote the dependence of $\eta$ on $x$. Let $f^\ast$ denote the additive approximation to $f$ having the maximum possible expected log-likelihood under the model. Maximum likelihood is used to fit an…

半鞅回归模型的筛法估计AI 解读

Estimation by the method of sieves for a semimartingale regression model introduced by Aalen (1980) is studied. It is of interest to estimate functions which describe the influence of the covariates over time. An estimator for these functions is introduced and conditions which ensure consistency of the estimator in $L^2$-norm are given. Applications to diffusion processes and point processes with censored data are also discussed.

关于平滑先验与非线性回归的一个注记AI 解读

A generalization of a nonparametric estimator proposed by Shiller (1984) is considered. The estimator is shown to be directly related to penalized least squares estimation and spline smoothing. Some simplifications concerning the computation of Shiller's estimator and corresponding posterior covariances are indicated.

天气与售电量关系的半参数估计AI 解读

A nonlinear relationship between electricity sales and temperature is estimated using a semiparametric regression procedure that easily allows linear transformations of the data. This accommodates introduction of covariates, timing adjustments due to the actual billing schedules, and serial correlation. The procedure is an extension of smoothing splines with the smoothness parameter estimated from minimization of the generalized cross-validation criterion introduced by…

右删失数据分位数函数的核型估计量AI 解读

Based on right-censored data from a lifetime distribution F 0, a kernel-type estimator of the quantile function Qo (p) = inf{t: F 0(t) ≧ p}, 0 ≦ p ≦ 1, is proposed. The estimator is defined by , which is smoother than the usual product-limit quantile function , where denotes the product-limit estimator of F 0 from the censored sample. Under the random censorship model and general conditions on hn, K, and F 0, it is shown that Qn (p) is strongly consistent. In addition, an…

部分线性模型中的样条平滑AI 解读

SUMMARY Suppose that Yi = X'iβ + f(ti) + εi, 1 ≤ i ≤ n, where β, f, and εi are unknown, but the m-th derivative of f is square integrable. Estimates of β and f are given which minimize the sum of the residual sum of squares and a roughness penalty. It is shown that these estimates are Bayes under a diffuse prior on β and f, and that, under mild conditions on Xi, ti, εi, and the roughness penalty, the estimate of β is consistent and asymptotically normal.

非线性回归中的残差方差与残差模式AI 解读

A nonparametric estimator of residual variance in nonlinear regression is proposed. It is based on local linear fitting. Asymptotically the estimator has a small bias, but a larger variance compared with the parametric estimator in linear regression. Finite sample properties are investigated in a simulation study, including a comparison with other nonparametric estimators. The method is also useful for spotting heteroscedasticity and outliers in the residuals at an early…

离散判别分析中核密度估计的另一种平滑参数选择AI 解读

The kernel method of estimating the cell probabilities of a multivariate categorical distribution, due to Aitchison & Aitken (1976), depends crucially on an unknown smoothing parameter λ. A method of estimating A is introduced which is explicitly connected to multivariate discrimination. The method, based on maximization of the leaving-one-out estimator of the nonerror rate, is shown to be Bayes risk strongly consistent. An example is given to illustrate the application.

两部关于密度估计的著作AI 解读

here K is a kernel function (usually a symmetric probability density function) and h is a smoothing parameter or bandwidth, the value of which determines how much the data are smoothed to produce the density estimate. The first published paper specifically on density estimation was Rosenblatt (1956); however, density estimates were suggested several years earlier by Fix and Hodges (1951) in a technical report. It is regrettable that this report was never published, since it…

非参数回归函数估计中的最优带宽选择AI 解读

Kernel estimators of an unknown multivariate regression function are investigated. A bandwidth-selection rule is considered, which can be formulated in terms of cross validation. Under mild assumptions on the kernel and the unknown regression function, it is seen that this rule is asymptotically optimal.

广义样条平滑中选择平滑参数的 GCV 与 GML 比较AI 解读

The partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood (GML) estimate for the smoothing parameter. Then this estimate is compared with the generalized cross validation (GCV) estimate both analytically and by Monte Carlo methods. The comparison is based on a predictive mean square error criteria. It is shown that if the true, unknown function being estimated is smooth in a sense to be defined then the GML…

从 Stein 无偏风险估计到广义交叉验证AI 解读

This paper concerns the method of generalized cross validation (GCV), a promising way of choosing between linear estimates. Based on Stein estimates and the associated unbiased risk estimates (Stein, 1981), a new approach to GCV is developed. Many consistency results are obtained for the cross-validated (Steinized) estimates in the contexts of nearest-neighbor nonparametric regression, model selection, ridge regression, and smoothing splines. Moreover, the associated Stein's…

初始条件未完全指定的状态空间模型：估计、滤波与平滑AI 解读

The likelihood is defined for a state space model with incompletely specified initial conditions by transforming the data to eliminate the dependence on the unspecified conditions. This approach is extended to obtain estimates of the state vectors and predictors and interpolators for missing observations. It is then shown that this method is equivalent to placing a diffuse prior distribution on the unspecified part of the initial state vector, and modified versions of the…

分位数函数的平滑非参数估计AI 解读

A smooth alternative to the conventional sample quantile function as a nonparametric estimator of a population quantile function is proposed. The proposed estimator is essentially a kernel type of estimator and has the same asymptotic distribution as the conventional sample quantile. The mean squared convergence rate of the proposed estimator is also estimated. Monte Carlo studies are conducted to compare the proposed estimator with the sample quantile and the estimator…

非参数回归曲线拟合中样条平滑方法的若干方面AI 解读

SUMMARY Non-parametric regression using cubic splines is an attractive, flexible and widely-applicable approach to curve estimation. Although the basic idea was formulated many years ago, the method is not as widely known or adopted as perhaps it should be. The topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology. Particular subjects covered include the basic principles of the…

分布尾指数的核估计AI 解读

We introduce a new estimate of the exponent of a distribution whose tail varies regularly at infinity. This estimate is expressed as the convolution of a kernel with the logarithm of the quantile function, and includes as particular cases the estimates introduced by Hill and by De Haan. Under very weak conditions, we prove asymptotic normality, consistency and discuss the optimal choices of the kernel and of the bandwidth parameter.

变带宽核估计的 L1 一致性札记AI 解读

A sample $X_1, \cdots, X_n$ of i.i.d. $R^d$-valued random vectors with common density $f$ is used to construct the density estimate $f_n(x) = (1/n) \sum^n_{i = 1} H^{-d}_{ni}K((x - X_i)/H_{ni}),$ where $K$ is a given density on $R^d$, and the $H_{ni}$'s are positive functions of $n, i$ and $X_1, \cdots, X_n$ (but not of $x$). The $H_{ni}$'s can be thought of as locally adapted smoothing parameters. We give sufficient conditions for the weak convergence to 0 of $\int |f_n - f|$…

核密度估计带宽选择的渐近有效解法AI 解读

A data-driven method of choosing the bandwidth, $h$, of a kernel density estimator is heuristically motivated by considering modifications of the Kullback-Leibler or pseudo-likelihood cross-validation function. It is seen that this means of choosing $h$ is asymptotically equivalent to taking the $h$ that minimizes some compelling error criteria such as the average squared error and the integrated squared error. Thus, for a given kernel function, the bandwidth can be chosen…

回归模型中的样条平滑与L₂渐近效率AI 解读

For nonparametric regression estimation on a bounded interval, optimal rates of decrease for integrated mean square error are known but not the best possible constants. A sharp result on such a constant, i.e., an analog of Fisher's bound for asymptotic variances is obtained for minimax risk over a Sobolev smoothness class. Normality of errors is assumed. The method is based on applying a recent result on minimax filtering in Hilbert space. A variant of spline smoothing is…

样条平滑与非参数回归的最优收敛速率AI 解读

Linear estimation is considered in nonparametric regression models of the form $Y_i = f(x_i) + \varepsilon_i, x_i \in (a, b)$, where the zero mean errors are uncorrelated with common variance $\sigma^2$ and the response function $f$ is assumed only to have a bounded square integrable $q$th derivative. The linear estimator which minimizes the maximum mean squared error summed over the observation points is derived, and the exact minimax rate of convergence is obtained. For…

平均平移直方图：有效的多维非参数密度估计量AI 解读

We introduce two nonparametric multivariate density estimators that are particularly suitable for application in interactive computing environments. These estimators are statistically comparable to kernel methods and computationally comparable to histogram methods. Asymptotic theory of the estimators is presented and examples with univariate and simulated trivariate Gaussian data are illustrated.