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病例对照研究的最优序贯设计AI 解读

Fixed case-control studies separately collect a case sample and a control sample withthe two sample sizes being fixed prior to studies and sometimes arbitrarily chosen. This often results in loss of efficiency of case-control designs in terms of cost-saving or time-saving of the studies. We study sequential case-control designs and, in connection with treatment allocation and stochastic approximation, derive a simple sampling rule that leads to optimal case-control designs.…

GARCH(1,1) 过程样本自相关与极值的极限理论AI 解读

The asymptotic theory for the sample autocorrelations and extremes of a GARCH (1, 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to 1, that is, when one is close to an infinite variance marginal distribution. This situation has been observed for various financial log-return series and led to the introduction of the IGARCH model. In such a situation, the sample autocorrelations are unreliable estimators of…

基于同胞配对数据定位数量性状位点的多元回归理论分析AI 解读

The interval mapping method has been shown to be a powerful tool for mapping QTL. However, it is still a challenge to perform a simultaneous analysis of several linked QTLs, and to isolate multiple linked QTLs. To circumvent these problems, multiple regression analysis has been suggested for experimental species. In this paper, the multiple regression approach is extended to human sib-pair data through multiple regression of the squared difference in trait values between two…

Lasso 型估计量的渐近性质AI 解读

We consider the asymptotic behavior ofregression estimators that minimize the residual sum of squares plus a penalty proportional to $\sum|\beta_j|^{\gamma}$. for some $\gamma > 0$. These estimators include the Lasso as a special case when $\gamma = 1$. Under appropriate conditions, we show that the limiting distributions can have positive probability mass at 0 when the true value of the parameter is 0.We also consider asymptotics for “nearly singular” designs.

通过模型选择自适应估计二次泛函AI 解读

We consider the problem of estimating $\|s\|^2$ when $s$ belongs to some separable Hilbert space and one observes the Gaussian process $Y(t) = \langles, t\rangle + \sigmaL(t)$, for all $t \epsilon \mathbb{H}$,where $L$ is some Gaussian isonormal process. This framework allows us in particular to consider the classical “Gaussian sequence model” for which $\mathbb{H} = l_2(\mathbb{N}*)$ and $L(t) = \sum_{\lambda\geq1}t_{\lambda}\varepsilon_{\lambda}$, where $(\varepsilon_{\lambda})_{\lambda\geq1}$…

正脉冲序列的参数反卷积AI 解读

This paper describes a parametric deconvolution method (PDPS) appropriate for a particular class of signals which we call spike-convolution models. These models arise when a sparse spike train —Dirac deltas according to our mathematical treatment —is convolved with a fixed point-spread function, and additive noise or measurement error is superimposed.We view deconvolution as an estimation problem, regarding the locations and heights of the underlying spikes, as well as the…

含隐藏变量的条件独立树：几何、矩与识别AI 解读

We study the geometry of the parameter space for Bayesian directed graphical models with hidden variables that have a tree structure and where all the nodes are binary.We show that the conditional independence statements implicit in such models can be expressed in terms of polynomial relationships among the central moments.This algebraic structure will enable us to identify the inequality constraints on the space of the manifest variables that are induced by the conditional…

二元变量分布中多元全正性的似然比检验AI 解读

Multivariate Totally Positive $(MTP_2)$ binary distributions have been studied in many fields, such as statistical mechanics, computer storage and latent variable models. We show that $MTP_2$ is equivalent to the requirement that the parameters of a saturated log-linear model belong to a convex cone, and we provide a Fisher-scoring algorithm for maximum likelihood estimation.We also show that the asymptotic distribution of the log-likelihood ratio is a mixture of chi-squares…

高斯混合筛估计的收敛速度AI 解读

Gaussian mixtures provide a convenient method of density estimation that lies somewhere between parametric models and kernel density estimators.When the number of components of the mixture is allowed to increase as sample size increases, the model is called a mixture sieve.We establish a bound on the rate of convergence in Hellinger distance for density estimation using the Gaussian mixture sieve assuming that the true density is itself a mixture of Gaussians; the underlying…

调查抽样中的局部多项式回归估计AI 解读

Estimation of finite population totals in the presence of auxiliary information is considered. A class of estimators based on local polynomial regression is proposed. Like generalized regression estimators, these estimators are weighted linear combinations of study variables, in which the weights are calibrated to known control totals, but the assumptions on the superpopulation model are considerably weaker. The estimators are shown to be asymptotically design-unbiased and…

纵向数据的非参数动态加性回归模型AI 解读

In this work we study additive dynamic regression models for longitudinal data. These models provide a flexible and nonparametric method for investigating the time-dynamics of longitudinal data. The methodology is aimed at data where measurements are recorded at random time points. We model the conditional mean of responses given the full internal history and possibly time-varying covariates. We derive the asymptotic distribution for a new nonparametric least squares…

比例风险回归的函数型 ANOVA 建模AI 解读

The logarithm of the relative risk function in a proportional hazards model involving one or more possibly time-dependent covariates is treated as a specified sum of a constant term, main effects, and selected interaction terms. Maximum partial likelihood estimation is used, where the maximization is taken over a suitably chosen finite-dimensional estimation space, whose dimension increases with the sample size and which is constructed from linear spaces of functions of one…

形状约束回归中的自由度AI 解读

For the problem of estimating a regression function, $\mu$ say, subject to shape constraints, like monotonicity or convexity, it is argued that the divergence of the maximum likelihood estimator provides a useful measure of the effective dimension of the model. Inequalities are derived for the expected mean squared error of the maximum likelihood estimator and the expected residual sum of squares. These generalize equalities from the case of linear regression. As an…

两条 Gibbs 采样链的对偶耦合AI 解读

Two coupled Gibbs sampler chains, both with invariant probability density $p$, are run in parallel so that the chains are negatively correlated.We define an asymptotically unbiased estimator of the $\pi$-expectation $E(f(\mathbf(X))$ which achieves significant variance reduction with respect to the usual Gibbs sampler at comparable computational cost. The variance of the estimator based on the new algorithm is always smaller than the variance of a single Gibbs sampler chain,…

回归函数单调性的检验AI 解读

We consider the problem of testing monotonicity of the regression function in a nonparametric regression model. We introduce test statistics that are functionals of a certain natural $U$-process. We study the limiting distribution of these test statistics through strong approximation methods and the extreme value theory for Gaussian processes. We show that the tests are consistent against general alternatives.

单调平均剩余寿命的估计AI 解读

In survival analysis and in the analysis of life tables an important biometric function of interest is the life expectancy at age $x, M(x)$, defined by $$M(x) = E[X - x|X > x],$$ where $X$ is a lifetime. $M$ is called the mean residual life function.In many applications it is reasonable to assume that $M$ is decreasing (DMRL) or increasing (IMRL); we write decreasing (increasing) for nonincreasing (non-decreasing). There is some literature on empirical estimators of $M$ and…

非参数回归中的函数聚合AI 解读

We consider the problem of estimating an unknown function $f$ from $N$ noisy observations on a random grid. In this paper we address the following aggregation problem: given $M$ functions $f_1,\dots, f_M$, find an “aggregated ”estimator which approximates $f$ nearly as well as the best convex combination $f^*$ of $f_1,\dots, f_M$. We propose algorithms which provide approximations of $f^*$ with expected $L_2$ accuracy $O(N^{-1/4}\ln^{1/4} M$. We show that this approximation…

约束局部 M 估计量的渐近性质AI 解读

We discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and,more generally, $M$-estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set $S$.We give a counterexample showing that regularity of $S$ in the sense of Clarke is not sufficient for asymptotic equivalence of $\sqrt{n}$-consistent locally optimal $M$-estimators.We argue further that stronger properties, such as so-called near…

平滑单调密度与单峰密度的极大似然估计AI 解读

We study the nonparametric estimation of univariate monotone and unimodal densities usingthe maximum smoothed likelihood approach. The monotone estimator is the derivative of the least concave majorant of the distribution correspondingto a kernel estimator.We prove that the mapping on distributions $\Phi$ with density $\varphi$, $$\varphi \mapsto \text{the derivative of the least concave majorant of $\Phi},$$ is a contraction in all $L^P$ norms $(1 \leq p \leq \infty)$, and…

非线性随机回归模型中的强一致性AI 解读

The class of nonlinear stochastic regression models includes most of the linear and nonlinear models used in time series, stochastic control and stochastic approximation schemes.The consistency of least squares estimators was established first by Lai.We present another set of sufficient conditions for consistency, which avoid the use of partial derivatives and are closer in spirit to the conditions presented by Wu for non-stochastic regression models with independent errors.

随时间漂移的自回归参数的递归估计AI 解读

Suppose the $X_0,\dots, X_n$ are observations of a one-dimensional stochastic dynamic process described by autoregression equations when the autoregressive parameter is drifted with time, i.e. it is some function of time: $\theta_0,\dots, \theta_n$, with $\theta_k = \theta(k/n)$. The function $\theta(t)$ is assumed to belong a priori to a predetermined nonparametric class of functions satisfying the Lipschitz smoothness condition. At each time point $t$ those observations…

泊松过程的序贯检验问题AI 解读

We present the explicit solution of the Bayesian problem of sequential testing of two simple hypotheses about the intensity of an observed Poisson process. The method of proof consists of reducing the initial problem to a free-boundary differential-difference Stephan problem and solving the latter by use of the principles of smooth and continuous fit. A rigorous proof of the optimality of the Wald’s sequential probability ratio test in the variational formulation of the…

非参数扩散模型的自适应漂移估计AI 解读

We consider a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable. The goal is to estimate the unknown drift coefficient.We apply a locally linear smoother with a data-driven bandwidth choice. The procedure is fully adaptive and nearly optimal up to a log log factor. The results about the quality of estimation are nonasymptotic and do not require any ergodic or mixing properties of the observed process.

面板计数数据下计数过程均值的两种估计量AI 解读

We study two estimators of the mean function of a countingprocess based on “panel count data.” The setting for “panel count data” is one in which $n$ independent subjects, each with a counting process with common mean function, are observed at several possibly different times duringa study. Following a model proposed by Schick and Yu, we allow the number of observation times, and the observation times themselves, to be random variables. Our goal is to estimate the mean…