统计学-部分经典文献

Estimation in Two Classes of Semiparametric Diffusion Models

两类半参数扩散模型的估计

In this paper we propose an estimation method for two classes of semiparametric scalar diffusion models driven by a Brownian motion: In the first class, only the diffusion term is parameterised while the drift is unspecified; in the second, the drift term is specifified while the diffffusion is of unknown form. The estimation method is based on the assumption of stationarity of the observed process. This allows us to express the unspecified term as a functional of the parameteric part and the stationary density. A MLE-like estimator for the parametric part and a kernel estimator of the nonparametric part are defined for a discrete sample with a fixed time distance between the observations. We show that the parametric part of the estimator is n consistent while the nonparametric part has a slower convergence rate. Also, the asymptotic distribution of the estimator is derived. To illustrate the usefulness of these two classes, we fit a specific model from the first class to a proxy of the Eurodollar short-term interest rate. We find non-linearities in both the drift and diffusion function that standard parametric models are unable to capture.

本文提出了两类半参数标量的估计方法由布朗运动驱动的扩散模型：在第一类中，只有扩散项参数化，而漂移是未指定的；在第二种情况下，漂移项被指定，而扩散项形式未知。估计方法是基于的假设观测过程的平稳性。这允许我们将未指定项表示为参数部分的函数和稳态密度。对于有固定观测时间距离的离散样本，定义了参数部分的极大似然估计量和非参数部分的核估计量。我们证明了参

数部分估计量是n一致的，而非参数部分收敛速度较慢。同时，给出了估计量的渐近分布。为了说明这两个类的有用性，我们将第一个类拟合一个特定模型与欧洲美元短期利率的代表相匹配。我们发现，漂移和扩散函数的非线性标准参数模型无法捕获。

Statistical estimation in variable coefficient models

变系数模型的统计估计

Varying coefficient models are a useful extension of classical linear models. They arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age. The appeal of these models is that the coefficient functions can easily be estimated via a simple local regression. This yields a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when different coefficient functions admit different degrees of smoothness. This drawback can be repaired by using our proposed two-step estimation procedure. The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an application to an environmental data set.

变系数模型是经典线性模型的一个有用的推广模型。当一个人想要研究回归系数在不同的群体中是如何变化的，并以特定的协变量(如年龄)为特征时，回归系数就会自然地出现。这些模型的优点是通过简单的局部回归就可以很容易地估计出系数函数。这产生了一个简单的一步估计过程。我们证明，当不同的系数函数有不同的平滑度时，这种一步法不可能是最优的。这个缺陷可以通过使用我们