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二元反应生物测定中的核估计与 Probit 估计AI 解读

A kernel method for the estimation of quantal dose-response curves is considered. In contrast to parametric modeling, this local smoothing method does not require any assumptions beyond smoothness of the dose-response curve and, in this sense, is nonparametric. In finite-sample situations, the kernel estimate of the dose-response curve is not necessarily monotone and, therefore, if the additional assumption of monotonicity of the dose-response curve is made, a monotonized…

摘要暂缺

部分线性模型中的核平滑AI 解读

SUMMARY Kernel smoothing is studied in partial linear models, i.e. semiparametric models of the form yi=ξi′β+f(ti)+εi(1⩽i⩽n), where the ξi are fixed known p vectors, β is an unknown vector parameter and f is a smooth but unknown function. Two methods of estimating β and f are considered, one related to partial smoothing splines and the other motivated by partial residual analysis. Under suitable assumptions, the asymptotic bias and variance are obtained for both methods, and…

高斯过程协方差的筛估计量AI 解读

Maximum likelihood estimation for the covariance $R$ of a zero-mean Gaussian process is considered, with no assumptions on the covariance or the "time" parameter set $T$. It is shown that the likelihood function is a.s. unbounded in general, and a sieve estimator $\hat{R}$ is constructed. The distribution of $\hat{R}$, considered as a process on $T \times T$, can be described exactly if a certain technical assumption is satisfied concerning the bivariate series expansion of $R$…

核密度众数估计的弱收敛与最优性AI 解读

A mode of a probability density $f(t)$ is a value $\theta$ that maximizes $f$. The problem of estimating the location of the mode is considered here. Estimates of the mode are considered via kernel density estimates. Previous results on this problem have several serious drawbacks. Conditions on the underlying density $f$ are imposed globally (rather than locally in a neighborhood of $\theta$). Moreover, fixed bandwidth sequences are considered, resulting in an estimate of…

用于贝叶斯非参数预测密度的逻辑斯蒂正态分布AI 解读

This article models the common density of an exchangeable sequence of observations by a generalization of the process derived from a logistic transform of a Gaussian process. The support of the logistic normal includes all distributions that are absolutely continuous with respect to the dominating measure of the observations. The logistic-normal family is closed in the prior to posterior Bayes analysis, with the observations entering the posterior distribution through the…

同时估计尺度曲线的稳健非参数回归AI 解读

Let $\{X_i, Y_i\}^n_{i=1} \subset \mathbb{R}^d \times \mathbb{R}$ be independent identically distributed random variables. If the conditional distribution $F(y \mid x)$ can be parametrized by $F(y \mid x) = F_0((y - m(x))/\sigma(x))$ with a fixed and known distribution $F_0$, the regression curve $m(x)$ and scale curve $\sigma(x)$ could be estimated by some parametric method. More generally, we assume that $F$ is unknown and consider nonparametric simultaneous $M$-type…

半参数部分样条与广义样条模型中的参数零模型检验AI 解读

Cox and Koh (1986) considered the model $y_i = f(x(i)) + \varepsilon_i, \varepsilon_i$ i.i.d. $N(0, \sigma^2)$, with the (parametric) null hypothesis $f(x), x \in \lbrack 0, 1 \rbrack$, a polynomial of degree $m - 1$ or less, versus the alternative $f$ is "smooth," based on the Bayesian model for $f$ which leads to polynomial smoothing spline estimates for $f$. They showed that there was no uniformly most powerful test and found the locally most powerful (LMP) test. We…

密度估计中的核阶数选择AI 解读

The selection of the order, i.e., number of vanishing moments, of the kernel in a kernel density estimator is considered from two points of view. First, theoretical properties are investigated by a mean integrated squared error analysis of the problem. Second, and perhaps more importantly, cross validation is proposed as a practical method of choice, and theoretical backing for this is provided through an asymptotic optimality result.

非参数回归中的自助法：局部自适应平滑与置信带AI 解读

The operation of the bootstrap in the context of nonparametric regression is considered. Bootstrap samples are taken from estimated residuals to study the distribution of a suitably recentered kernel estimator. The application of this principle to the problem of local adaptive choice of bandwidth and to the construction of confidence bands is investigated and compared with a direct method based on asymptotic means and variances. The technique of the bootstrap is to replace…

自动选择的回归平滑参数距最优值有多远？AI 解读

We address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation. Call the “optimal bandwidth” the minimizer of the average squared error. We consider several automatically selected bandwidths that approximate the optimum. How far are the automatically selected bandwidths from the optimum? The answer is studied theoretically and through simulations. The theoretical results include a central…

随机右删失样本核密度估计的渐近最优带宽选择AI 解读

This paper makes two important contributions to the theory of bandwidth selection for kernel density estimators under right censorship. First, an asymptotic representation of the integrated squared error into easily understood variance and squared bias components is given. Second, it is shown that if the bandwidth is chosen by the data-based method of least-squares cross-validation, then it is asymptotically optimal in a compelling sense. A by-product of the first part is an…

Kullback—Leibler损失与密度估计AI 解读

"Discrimination information," or Kullback-Leibler loss, is an appropriate measure of distance in problems of discrimination. We examine it in the context of nonparametric kernel density estimation and show that its asymptotic properties are profoundly influenced by tail properties of the kernel and of the unknown density. We suggest ways of choosing the kernel so as to reduce loss, and describe the extent to which likelihood cross-validation asymptotically minimises loss.…

密度估计中的有偏与无偏交叉验证AI 解读

Nonparametric density estimation requires the specification of smoothing parameters. The demands of statistical objectivity make it highly desirable to base the choice on properties of the data set. In this article we introduce some biased cross-validation criteria for selection of smoothing parameters for kernel and histogram density estimators, closely related to one investigated in Scott and Factor (1981). These criteria are obtained by estimating L 2 norms of derivatives…

球面数据的核密度估计AI 解读

We study two natural classes of kernel density estimators for use with spherical data. Members of both classes have already been used in practice. The classes have an element in common, but for the most part they are disjoint. However, all members of the first class are asymptotically equivalent to one another, and to a single element of the second class. In this sense the second class ‘contains’ the first. It includes some estimators which out-perform all those in the first…

摘要暂缺

Efron–Stein不等式在密度估计中的应用AI 解读

The Efron-Stein inequality is applied to prove that the kernel density estimate $f_n$, with an arbitrary nonnegative kernel and an arbitrary smoothing factor, satisfies the inequality $\operatorname{var}(\int|f_n - f|) \leq 4/n$ for all densities $f$. Similar inequalities are obtained for other estimates.

密度估计的半参数方法AI 解读

One method of fitting a parametric density function f(x, θ) is first to estimate θ by maximum likelihood, say, and then to estimate f(x, θ) by . On the other hand, when the parametric model does not hold, the true density f(x) may be estimated nonparametrically, as in the case of a kernel estimate . The key idea proposed is to fit a combination of the parametric and nonparametric estimates, namely where π (0 ≤ π ≤ 1) is unknown. The parameter π is estimated from the data,…

未知形式异方差下的渐近有效估计AI 解读

In a multiple-regression model the residual variance is an unknown function of the explanatory variables, and estimated by nearest-neighbor nonparametric regression. The resulting weighted least-squares estimator of the regression coefficients is shown to be adaptive, in the sense of having the same asymptotic distribution, to first order, as estimators based on knowledge of the actual variance function or a finite parameterization of it. A similar result was established by…

回归分析中的异方差估计AI 解读

Consider the regression model $Y_i = g(t_i) + \varepsilon_i, 1 \leq i \leq n$, with nonrandom design variables $(t_i)$ and measurements $(Y_i)$ for the unknown regression function $g(\cdot)$. We assume that the data are heteroscedastic, i.e., $E(\varepsilon^2_i) = \sigma^2_i \not\equiv \operatorname{const.}$ and investigate how to estimate $\sigma^2_i$. If $\sigma^2_i = \sigma^2(t_i)$ with a smooth function $\sigma^2(\cdot)$, initial estimators $\tilde{\sigma}^2_i$ can be…

使用估计得分函数的线性秩统计量的局部渐近分析AI 解读

Modified versions of linear rank statistics $S_N(\varphi)$ with score function $\varphi$ are studied for the two-sample testing problem of randomness. Depending on the unknown underlying alternatives, some score function $\varphi = b$ is known to be approximately optimal. Behnen, Neuhaus and Ruymgaart (1983) proposed estimating $b$ by some estimator $\hat b_N$ obtained by the kernel method based on ranks and used the quadratic rank statistic $S_N(\hat{b}_N)$ for testing the…

乘法强度模型中的筛法极大似然估计AI 解读

For point processes comprising i.i.d. copies of a multiplicative intensity process, it is shown that even though log-likelihood functions are unbounded, consistent maximum likelihood estimators of the unknown function in the stochastic intensity can be constructed using the method of sieves. Conditions are given for existence and strong and weak consistency, in the $L^1$-norm, of suitably defined maximum likelihood estimators. A theorem on local asymptotic normality of…

分支曲线的非参数估计AI 解读

The spline smoothing approach to nonparametric regression is extended to cover the estimation of curves that branch at certain specified points in time. This extension is motivated and illustrated by data from an experiment on sunflowers. Apart from providing insight into this particular data-analytic problem, the work described demonstrates the power of the roughness penalty approach in coping with curve estimation of a nonstandard kind. Particularly in the context of…

回归曲线的可变带宽核估计AI 解读

In the model $Y_i = g(t_i) + \varepsilon_i,\quad i = 1,\cdots, n,$ where $Y_i$ are given observations, $\varepsilon_i$ i.i.d. noise variables and $t_i$ nonrandom design points, kernel estimators for the regression function $g(t)$ with variable bandwidth (smoothing parameter) depending on $t$ are proposed. It is shown that in terms of asymptotic integrated mean squared error, kernel estimators with such a local bandwidth choice are superior to the ordinary kernel estimators…