Convergence of normalized quadratic forms

The asymptotic behavior of quadratic forms of stationary sequences plays an important role in statistics, for example, in the context of the Whittle approximation to maximum likelihood. The quadratic form, appropriately normalized, may have Gaussian or non-Gaussian limits. Under what circumstances will the limits be of one type or another? And if the limits are non-Gaussian, what are they? The goal of this paper is to describe the historical development of the problem and provide further extensions of recent results.     

Minimum Sample Size Considerations for Two-Group Linear and Quadratic Discriminant Analysis with Rare Populations

Linear discriminant analysis and quadratic discriminant analysis are used to predict group membership. Rare populations present situations in which group sizes differ drastically. This article examined k = 2 and k = 4 predictor variables for groups with different levels of rarity and different levels of sensitivity and specificity. Sample size recommendations were generated for both minimum and maximum group overlap using the leave-one-out (L-O-O) method of estimation. Minimum sample size recommendations are provided in tables for immediate implementation by applied researchers.     

Robust nonparametric estimation of the intensity function of point data

Intensity functions—which describe the spatial distribution of the occurrences of point processes—are useful for risk assessment. This paper deals with the robust nonparametric estimation of the intensity function of space–time data from events such as earthquakes. The basic approach consists of smoothing the frequency histograms with the local polynomial regression (LPR) estimator. This method allows for automatic boundary corrections, and its jump-preserving ability can be improved with robustness. We derive a robust local smoother from the weighted-average approach to M-estimation and we select its bandwidths with robust cross-validation (RCV). Further, we develop a robust recursive algorithm for sequential processing of the data binned in time. An extensive application to the Northern California earthquake catalog in the San Francisco, CA, area illustrates the method and proves its validity.     

Ad hoc网络密钥预配置方案

为了减少Ad hoc网络密钥管理方案对网络资源、节点资源的依赖,同时提高网络的安全性能,该文深入研究了传感器网络中基于多项式的密钥对预配置方案,将门限机制引用到密钥的传输过程当中,设计出一个应用于Ad hoc网络的高效密钥预配置方案:基于门限机制的密钥预配置方案。通过分析可以看出该方案有一些优良的特性,包括两节点间能以很高的概率成功建立密钥对、对入侵有较强的鲁棒性、高连通性、低通讯量。     

Automatic Local Smoothing for Spectral Density Estimation

This article uses local polynomial techniques to fit Whittle's likelihood for spectral density estimation. Asymptotic sampling properties of the proposed estimators are derived, and adaptation of the proposed estimator to the boundary effect is demonstrated. We show that the Whittle likelihood-based estimator has advantages over the least-squares based log-periodogram. The bandwidth for the Whittle likelihood-based method is chosen by a simple adjustment of a bandwidth selector proposed in Fan & Gijbels (1995). The effectiveness of the proposed procedure is demonstrated by a few simulated and real numerical examples. Our simulation results support the asymptotic theory that the likelihood based spectral density and log-spectral density estimators are the most appealing among their peers     

On difference-based variance estimation in nonparametric regression when the covariate is high dimensional

Summary.  We consider the problem of estimating the noise variance in homoscedastic nonparametric regression models. For low dimensional covariates t  ∈  R ^d ,  d =1, 2, difference-based estimators have been investigated in a series of papers. For a given length of such an estimator, difference schemes which minimize the asymptotic mean-squared error can be computed for d =1 and d =2. However, from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias. We provide theoretical support for these findings. In particular, we show that with increasing dimension d this becomes more drastic. If d 4, these estimators even fail to be consistent. A different class of estimators is discussed which allow better control of the bias and remain consistent when d 4. These estimators are compared numerically with kernel-type estimators (which are asymptotically efficient), and some guidance is given about when their use becomes necessary.     

关于指数Diophantine方程（x^m-1）／（x-1）=y^5

该文证明了：方程（x^m-1）／（x-1）=y^2．x〉1。y〉1，m〉2，没有正整数解（x，y，m）可使m=4（mod5）且m是平方数．     

Facility Dispersion Problems Under Capacity and Cost Constraints

The MAX-MIN dispersion problem, which arises in the placement of undesirable facilities, involves selecting a specified number of sites among a set of potential sites so as to maximize the minimum distance between any pair of selected sites. We consider different versions of this dispersion problem where each potential site has an associated storage capacity and a storage cost. A typical problem in this context is to choose a subset of potential sites so that the total capacity of the chosen sites is at least a given value, the total storage cost is within the specified budget and the minimum distance between any pair of chosen sites is maximized. Since these constrained optimization problems are NP-hard in general, we consider whether there are efficient approximation algorithms for them with good performance guarantees. Our results include approximation algorithms for some versions, approximation schemes for some geometric versions and polynomial algorithms for special cases. We also present results that bring out the intrinsic difficulty of obtaining near-optimal solutions to some versions.     

Bayesian analysis of nonlinear and non-Gaussian state space models via multiple-try sampling methods

We develop in this paper three multiple-try blocking schemes for Bayesian analysis of nonlinear and non-Gaussian state space models. To reduce the correlations between successive iterates and to avoid getting trapped in a local maximum, we construct Markov chains by drawing state variables in blocks with multiple trial points. The first and second methods adopt autoregressive and independent kernels to produce the trial points, while the third method uses samples along suitable directions. Using the time series structure of the state space models, the three sampling schemes can be implemented efficiently. In our multimodal examples, the three multiple-try samplers are able to generate the desired posterior sample, whereas existing methods fail to do so.     

Asymptotics of cross-validated risk estimation in estimator selection and performance assessment

Risk estimation is an important statistical question for the purposes of selecting a good estimator (i.e., model selection) and assessing its performance (i.e., estimating generalization error). This article introduces a general framework for cross-validation and derives distributional properties of cross-validated risk estimators in the context of estimator selection and performance assessment. Arbitrary classes of estimators are considered, including density estimators and predictors for both continuous and polychotomous outcomes. Results are provided for general full data loss functions (e.g., absolute and squared error, indicator, negative log density). A broad definition of cross-validation is used in order to cover leave-one-out cross-validation, V-fold cross-validation, Monte Carlo cross-validation, and bootstrap procedures. For estimator selection, finite sample risk bounds are derived and applied to establish the asymptotic optimality of cross-validation, in the sense that a selector based on a cross-validated risk estimator performs asymptotically as well as an optimal oracle selector based on the risk under the true, unknown data generating distribution. The asymptotic results are derived under the assumption that the size of the validation sets converges to infinity and hence do not cover leave-one-out cross-validation. For performance assessment, cross-validated risk estimators are shown to be consistent and asymptotically linear for the risk under the true data generating distribution and confidence intervals are derived for this unknown risk. Unlike previously published results, the theorems derived in this and our related articles apply to general data generating distributions, loss functions (i.e., parameters), estimators, and cross-validation procedures.