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删失回归：局部线性近似及其应用AI 解读

Various statistical tools are available for modeling the relationship between response and covariate if the data are fully observable. In the situation of censored data, however, those tools are no longer directly applicable. This article provides an easily implemented methodology for modeling the association, based on censored data. The form of the regression relationship will be completely determined by the data; no assumptions are made about this form. Basic ideas behind…

含变量测量误差的非参数回归AI 解读

The effect of errors in variables in nonparametric regression estimation is examined. To account for errors in covariates, deconvolution is involved in the construction of a new class of kernel estimators. It is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution. In fact, there are two types of rates of convergence according to whether the error…

In this paper, a method for finding global minimax lower bounds is introduced. The idea is to adjust automatically the direction of a local one-dimensional subproblem at each location to the nearly hardest one, and to use locally the difficulty of the one-dimensional subproblem. This method has the advantages of being easily implemented and understood. The lower bound is then applied to nonparametric deconvolution to obtain the optimal rates of convergence for estimating a…

局部线性回归平滑器及其极小极大效率AI 解读

In this paper we introduce a smooth version of local linear regression estimators and address their advantages. The MSE and MISE of the estimators are computed explicitly. It turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency-they are not only efficient in rates but also nearly efficient in constant factors. In the nonparametric regression context, the asymptotic minimax lower bound is developed via the heuristic of…

带宽选择的最优可达常数AI 解读

For the data based choice of the bandwidth of a kernel density estimator, several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth. In particular the relative rate of convergence is the square root of the sample size, which is known to be the best possible. The point of this paper is to show how semiparametric arguments can be employed to calculate the best possible constant coefficient, that is, an analog of…

可变带宽与局部线性回归平滑器AI 解读

In this paper we introduce an appealing nonparametric method for estimating the mean regression function. The proposed method combines the ideas of local linear smoothers and variable bandwidth. Hence, it also inherits the advantages of both approaches. We give expressions for the conditional MSE and MISE of the estimator. Minimization of the MISE leads to an explicit formula for an optimal choice of the variable bandwidth. Moreover, the merits of considering a variable…

适应设计特征的非参数回归AI 解读

In this article we study the method of nonparametric regression based on a weighted local linear regression. This method has advantages over other popular kernel methods. Moreover, such a regression procedure has the ability of design adaptation: It adapts to both random and fixed designs, to both highly clustered and nearly uniform designs, and even to both interior and boundary points. It is shown that the local linear regression smoothers have high asymptotic efficiency…

二次泛函的估计AI 解读

We discuss the difficulties of estimating quadratic functionals based on observations $Y(t)$ from the white noise model $Y(t) = \int^t_0 f(u) du + \sigma W(t),\quad t \in \lbrack 0, 1\rbrack,$ where $W(t)$ is a standard Wiener process on $\lbrack 0, 1\rbrack$. The optimal rates of convergence (as $\sigma \rightarrow 0$) for estimating quadratic functionals under certain geometric constraints are found. Specifically, the optimal rates of estimating $\int^1_0\lbrack f^{(k)}(x)\rbrack^2 dx$…

非参数反卷积问题的最优收敛速度AI 解读

Deconvolution problems arise in a variety of situations in statistics. An interesting problem is to estimate the density $f$ of a random variable $X$ based on $n$ i.i.d. observations from $Y = X + \varepsilon$, where $\varepsilon$ is a measurement error with a known distribution. In this paper, the effect of errors in variables of nonparametric deconvolution is examined. Insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error…