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多项式回归的E最优设计AI 解读

E-optmal designs for the full mean parameter vector, and for many subsets in univariate polynomial regression models are determined. The derivation is based on the interplay between E-optimality and scalar optimality. The scalar parameter systems are obtained as transformations of the coefficient vector c of the Chebyshev polynomial.

损失估计中的可容许性结果AI 解读

In this paper we consider the problem of estimating the loss of point estimators and study admissibility of such estimators of loss. We adapt and extend the "unified admissibility proof" idea of Brown and Hwang to this problem. We first present the result in the Gaussian setup. We then generalize the procedure to general exponential family distributions and apply it to the Poisson distribution. The result for the gamma distribution is also stated. The role played by the…

系统发育推断中若干概率模型的不变量AI 解读

The so-called method of invariants is a technique in the field of molecular evolution for inferring phylogenetic relations among a number of species on the basis of nucleotide sequence data. An invariant is a polynomial function of the probability distribution defined by a stochastic model for the observed nucleotide sequence. This function has the special property that it is identically zero for one possible phylogeny and typically nonzero for another possible phylogeny.…

效率约束下的位置参数偏差稳健估计AI 解读

In 1964, P. Huber established the following minimax bias robustness result for estimating the location $\mu$ in the $\varepsilon$-contamination family $F(x)=(1-\varepsilon)\phi[(x-\mu)/s]+\varepsilon H(x)$, where $\phi$ is the standard normal distribution and H is an arbitrary distribution function: The median minimizes the maximum asymptotic bias among all translation equivariant estimates of location. However, the median efficiency of $2/\pi$ at the Gaussian model may be…

污染偏差的下界：全局极小极大与局部线性估计AI 解读

We study how robust estimators can be in parametric families, obtaining a lower bound on the contamination bias of an estimator that holds for a wide class of parametric families. This lower bound includes as a special case the bound used to establish that the median is bias minimax among location equivariant estimators, and it is tight or nearly tight in a variety of other settings such as scale estimation, discrete exponential families and multiple linear regression. The…

用于自助法的平衡重要性重抽样AI 解读

We show that the method of importance resampling, introduced by Vernon Johns and Anthony Davison, may be enhanced by balancing the resamples. It is demonstrated that "balanced importance resampling" improves on both "balanced uniform resampling" and "random importance resampling", from the viewpoint of statistical efficiency. Moreover, the range of applications for which efficient resampling methods may be applied is extended to include statistics which are smooth functions…

回归函数可加性的检验AI 解读

Observations $y_{ij}$ are made at points $(x_{1i},x_{2j})$ according to the model $y_{iy}=F(x_{1i},x_{2j})+e_{ij}$, where the $e_{ij}$ are independent normals with constant variance. In order to test that $F(x_1,x_2)$ is an additive function of $x_1$ and $x_2$, a likelihood ratio test is constructed comparing $F(x_1,x_2)=\mu+Z_1 (x_1)+Z_2(x_2)$ with $F(x_1,x_2)=\mu+Z_1(x_1)+Z_2(x_2)+Z(x_1,x_2)$, where $Z_1$, $Z_2$ are Brownian motions and Z is a Brownian sheet. The ratio of…

用于非参数曲线估计的一类新核函数AI 解读

We introduce a new class of variable kernels which depend on the smoothing parameter b through a simple scaling operation, and which have good integrated mean square error (IMSE) convergence properties. These kernels deform "automatically" near the boundary, eliminating boundary bias. Computational formulas are given for all orders of kernel in terms of exponentially damped sines and cosines. The kernel is a computationally convenient approximation to a certain Green's…

单指标模型中的最优平滑AI 解读

Single-index models generalize linear regression. They have applications to a variety of fields, such as discrete choice analysis in econometrics and dose response models in biometrics, where high-dimensional regression models are often employed. Single-index models are similar to the first step of projection pursuit regression, a dimension-reduction method. In both cases the orientation vector can be estimated root-n consistently, even if the unknown univariate function (or…

截断数据乘积极限估计量的几乎必然表示AI 解读

In the left-truncation model, one observes data $(X_i,Y_i)$ only when $Y_i\leq X_i$. Let F denote the marginal d.f. of $X_i$ , the variable of interest. The nonparametric MLE $\hat{F}_n$ of F aims at reconstructing F from truncated data. In this paper an almost sure representation of $\hat{F}_n$ is derived with improved error bounds on the one hand and under weaker distributional assumptions on the other hand.

Cox 模型中的非参数估计AI 解读

Nonparametric estimation of the relative risk in a generalized Cox model with multivariate time dependent covariates is considered. Estimation is based on a penalized partial likelihood. Using techniques from Andersen and Gill, and Cox and O'Sullivan, upper bounds on rate of convergence in a variety of norms are obtained. These upper bounds match the optimal rates available for linear nonparametric regression and density estimation. The results are uniform in the smoothing…

删失数据的贝叶斯自助法AI 解读

A Bayesian bootstrap for a censored data model is introduced. Its small sample distributional properties are discussed and found to be similar to Efron's bootstrap for censored data. In the absence of censoring, the Bayesian bootstrap for censored data reduces to Rubin's Bayesian bootstrap for complete data. A first-order large-sample theory is developed. This theory shows that both censored data bootstraps are consistent bootstraps for approximating the sampling…

删失回归数据秩估计的大样本研究AI 解读

Large sample approximations are developed to establish asymptotic linearity of the commonly used linear rank estimating functions, defined as stochastic integrals of counting processes over the whole line, for censored regression data. These approximations lead to asymptotic normality of the resulting rank estimators defined as solutions of the linear rank estimating equations. A second kind of approximations is also developed to show that the estimating functions can be…

统计泛函的可微性与 Jackknife 一致性AI 解读

In statistical applications the unknown parameter of interest can frequently be defined as a functional $\theta=T(F)$, where F is an unknown population. Statistical inferences about $\theta$ are usually made based on the statistic $T(F_n)$, where $F_n$ is the empirical distribution. Assessing $T(F_n)$ (as an estimator of $\theta$) or making large sample inferences usually requires a consistent estimator of the asymptotic variance of $T(F_n)$. Asymptotic behavior of the…

条件分布的渐近最优检验AI 解读

Let $(X_1,Y_1),\cdots,(X_n,Y_n)$ be independent replicates of the random vector $(X,Y)\in \mathbb{R}^{d+m}$, where X is $\mathbb{R}^d$-valued and Y is $\mathbb{R}^m$-valued. We assume that the conditional distribution $P(Y\in\cdot|X=x)=Q_\vartheta(\cdot)$ of Y given X = x is a member of a parametric family, where the parameter space $\Theta$ is an open subset of $\mathbb{R}^k$ with $0\in\Theta$. Under suitable regularity conditions we establish upper bounds for the power…

若干非参数最大似然估计量的 Hellinger 一致性AI 解读

Consider a class $\mathscr{P}={P_\theta:\theta\in\Theta}$ of probability measures on a measurable space $(\mathscr{X},\mathscr{A})$, dominated by a $\sigma$ -finite measure $\mu$. Let $f_\theta=dP_\theta/d_\mu$, $\theta\ in\Theta$, and let $\theta_n$ be a maximum likelihood estimator based on n independent observations from $P_{\theta_0}$, $\theta_0\in\Theta$. We use results from empirical process theory to obtain convergence for the Hellinger distance $h(f_{\hat{\theta}_n}, f_{\theta_0})$…

奇异问题中有信息无偏估计量的不存在性AI 解读

In many nonparametric problems, such as density estimation, nonparametric regression and so on, all the existing informative estimators are biased (asymptotic or finite sample). There has long been a suspicion that either informative unbiased estimators do not exist for such problems or they must be quite complicated. In this paper, we clarify the nonexistence of informative unbiased estimators in all singular problems both for fixed sample size and asymptotically (this…

门限自回归模型最小二乘估计的相合性与极限分布AI 解读

It is shown that, under some regularity conditions, the least squares estimator of a stationary ergodic threshold autoregressive model is strongly consistent. The limiting distribution of the least squares estimator is derived. It is shown that the estimator of the threshold parameter is N consistent and its limiting distribution is related to a compound Poisson Process.

通过多重划分交叉验证进行模型选择AI 解读

A natural extension of the simple leave-one-out cross validation (CV) method is to allow the deletion of more than one observations. In this article, several notions of the multifold cross validation (MCV) method have been discussed. In the context of variable selection under a linear regression model, we show that the delete-d MCV criterion is asymptotically equivalent to the well known FPE criterion. Two computationally more feasible methods, the r-fold cross validation…

高维线性模型的自助法与野生自助法AI 解读

In this paper two bootstrap procedures are considered for the estimation of the distribution of linear contrasts and of F-test statistics in high dimensional linear models. An asymptotic approach will be chosen where the dimension p of the model may increase for sample size $n\rightarrow\infty$. The range of validity will be compared for the normal approximation and for the bootstrap procedures. Furthermore, it will be argued that the rates of convergence are different for…

平滑样条密度估计：理论AI 解读

In this article, a class of penalized likelihood probability density estimators is proposed and studied. The true log density is assumed to be a member of a reproducing kernel Hilbert space on a finite domain, not necessarily univariate, and the estimator is defined as the unique unconstrained minimizer of a penalized log likelihood functional in such a space. Under mild conditions, the existence of the estimator and the rate of convergence of the estimator in terms of the…

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…

Kolmogorov–Smirnov 损失下最佳不变分布函数估计量的极小极大性AI 解读

For the invariant decision problem of estimating a continuous distribution function with the Kolmogorov-Smirnov loss, it is proved that the best invariant estimator is minimax.

被选择总体尺度参数估计的不可容许性结果AI 解读

Let $X_1, X_2, \ldots, X_k$ be $k$ independent gamma random variables with different scale parameters but with a common known shape parameter. Suppose the population corresponding to the largest $X_{(1)}$ [or the smallest $X_{(k)}$] observation is selected. The problem of estimating the scale parameter $\theta_{(1)}$ [or $\theta_{(k)}$] of the selected population is considered. We derive, using the method of differential inequalities, explicit estimators that dominate the…