Design Adaptive Nearest Neighbor Regression Estimation

This paper deals with nonparametric regression estimation under arbitrary sampling with an unknown distribution. The effect of the distribution of the design, which is a nuisance parameter, can be eliminated by conditioning. An upper bound for the conditional mean squared error of k−NN estimates leads us to consider an optimal number of neighbors, which is a random function of the sampling. The corresponding estimate can be used for nonasymptotic inference and is also consistent under a minimal recurrence condition. Some deterministic equivalents are found for the random rate of convergence of this optimal estimate, for deterministic and random designs with vanishing or diverging densities. The proposed estimate is rate optimal for standard designs.     

Efficient loop-check for KD 45 logic

We introduce a new sequent calculus for KD 45 logic. A loop-check technique is used to determine whether a sequent derivable or not. We concentrate ourselves on the efficiency of the loop-check technique used. The efficiency is obtained by making the loop-check to act locally (then we need to check only one or two current sequents), instead of a global loop-check used in known works as [3]. Moreover, the sequent calculus introduced uses a marked operator □ to integrate loop-check into an inference rule. Besides the efficient loop-check used, the sequent calculus introduced produces smaller derivation trees that reduce the derivation time. We also prove a lemma which determines the maximal number of modality rule applications in one branch of a tree. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 55–66, January–March, 2006.     

Estimating a Distribution Function for Censored Time Series Data

Consider a long term study, where a series of dependent and possibly censored failure times is observed. Suppose that the failure times have a common marginal distribution function, but they exhibit a mode of time series structure such as α-mixing. The inference on the marginal distribution function is of interest to us. The main results of this article show that, under some regularity conditions, the Kaplan–Meier estimator enjoys uniform consistency with rates, and a stochastic process generated by the Kaplan–Meier estimator converges weakly to a certain Gaussian process with a specified covariance structure. Finally, an estimator of the limiting variance of the Kaplan–Meier estimator is proposed and its consistency is established.     

Asymptotics and bootstrap for inverse Gaussian regression

This paper studies regression, where the reciprocal of the mean of a dependent variable is considered to be a linear function of the regressor variables, and the observations on the dependent variable are assumed to have an inverse Gaussian distribution. The large sample theory for the pseudo maximum likelihood estimators is available in the literature, only when the number of replications increase at a fixed rate. This is inadequate for many practical applications. This paper establishes consistency and derives the asymptotic distribution for the pseudo maximum likelihood estimators under very general conditions on the design points. This includes the case where the number of replications do not grow large, as well as the one where there are no replications. The bootstrap procedure for inference on the regression parameters is also investigated.Research supported in part by NSF Grant DMS-9208066.Research supported in part by NSERC of Canada.     

On inference concerning time-dependent queue performance: The M/G/1 example

This paper proposes easily-computed approximations to the finite-time expected waiting time for anM/G/1 system starting from an empty state. Both unsaturated (ρ<1) and saturated (ρ>1) conditions are considered. Numerical evidence is presented to indicate that the quality of the approximations is usefully good, especially when ease of computation is an issue. Further, the methodology is adapted to assess expected waiting time when inference must be made from a random sample of service times, and the decision is made to do so nonparametrically, i.e., without fitting a specific function. The results appear reasonable and potentially useful, and are not burdensome to obtain. The methodology investigated can also be applied to the variety of queueing models that are close siblings ofM/G/1: priority and breakdowns and “vacations” being examples. Of course other approximating and inferential options remain to be investigated.     

M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps

We study parametric inference for multidimensional stochastic differential equations with jumps from some discrete observations. We consider a case where the structure of jumps is mainly controlled by a random measure which is generated by a Lévy process with a Lévy measure f_θ(z)dz, and we admit the case ∫ f_θ(z)dz = ∞ in which infinitely many small jumps occur even in any finite time intervals. We propose an estimating function under this complicated situation and show the consistency and the asymptotic normality. Although the estimators in this paper are not completely efficient, the method can be applied to comparatively wide class of stochastic differential equations, and it is easy to compute the estimating equations. Therefore, it may be useful in applications. We also present some simulation results for some simple models. Final version 25 December 2004     

基于两阶段思想处理拒绝推断的信用评分模型

本文首先从数据缺失机制的角度分析了信用评分模型的开发和应用中所存在的样本偏差问题,提出了可以用拒绝推断来处理此类问题;然后在曾经被应用于拒绝推断问题处理的Heckman两阶段模型的基础上,提出了用拟似然两阶段模型和广义偏线性模型这两种新的两阶段方法来处理信用评分模型中的拒绝推断问题。经过实证分析发现,应用这两种方法可以得到很理想的结果。另外根据本文的研究,人行征信这类外部数据是拒绝推断最有效的方法,如果此类数据缺乏,则用拟似然两阶段模型和广义偏线性模型是比较有效的拒绝推断方法。     

Unbiased multi-index Monte Carlo

We introduce a new class of Monte Carlo-based approximations of expectations of random variables such that their laws are only available via certain discretizations. Sampling from the discretized versions of these laws can typically introduce a bias. In this paper, we show how to remove that bias, by introducing a new version of multi-index Monte Carlo (MIMC) that has the added advantage of reducing the computational effort, relative to i.i.d. sampling from the most precise discretization, for a given level of error. We cover extensions of results regarding variance and optimality criteria for the new approach. We apply the methodology to the problem of computing an unbiased mollified version of the solution of a partial differential equation with random coefficients. A second application concerns the Bayesian inference (the smoothing problem) of an infinite-dimensional signal modeled by the solution of a stochastic partial differential equation that is observed on a discrete space grid and at discrete times. Both applications are complemented by numerical simulations.     

A Unified Approach to Second Order Optimality Criteria in Nonlinear Design of Experiments

In the nonlinear regression model we consider the optimal design problem with a second order design D-criterion. Our purpose is to present a general approach to this problem, which includes the asymptotic second order bias and variance criterion of the least squares estimator and criteria using the volume of confidence regions based on different statistics. Under assumptions of regularity for these statistics a second order approximation of the volume of these regions is derived which is proposed as a quadratic optimality criterion. These criteria include volumes of confidence regions based on the u _n - representable statistics. An important difference between the criteria presented in this paper and the second order criteria commonly employed in the recent literature is that the former criteria are independent of the vector of residuals. Moreover, a refined version of the commonly applied criteria is obtained, which also includes effects of nonlinearity caused by third derivatives of the response function.     

A new smoothing Newton-type method for second-order cone programming problems

A new smoothing function of the well-known Fischer–Burmeister function is given. Based on this new function, a smoothing Newton-type method is proposed for solving second-order cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Numerical results demonstrate the effectiveness of the algorithm.     

Root n bandwidths selectors in multivariate kernel density estimation

Based on a random sample of size n from an unknown d-dimensional density f, the problem of selecting the bandwidths in kernel estimation of f is investigated. The optimal root n relative convergence rate for bandwidth selection is established and the information bounds in this convergence are given, and a stabilized bandwidth selector (SBS) is proposed. It is known that for all d the bandwidths selected by the least squares cross-validation (LSCV) have large sample variations. The proposed SBS, as an improvement of LSCV, will reduce the variation of LSCV without significantly inflating its bias. The key idea of the SBS is to modify the d-dimensional sample characteristic function beyond some cut-off frequency in estimating the integrated squared bias. It is shown that for all d and sufficiently smooth f and kernel, if the bandwidth in each coordinate direction varies freely, then the multivariate SBS is asymptotically normal with the optimal root n relative convergence rate and achieves the (conjectured) lower bound on the covariance matrix.Part of the research was done while the first author was visiting the Institute of Statistical Science, Academia Sinica, Taipei, Taiwan. This work was supported by grant NSC-89-2118-M-006-011, NSC-90-2118-M-006-013 and NSC-91-2118-M-006-005 of National Science Council of Taiwan, R.O.C.     

A generalized Henstock-Stieltjes integral involving division functions

We can consider the Riemann-Stieltjes integral dg as an integral of a point function f with respect to an interval function g. We could extend it to the Henstock-Stieltjes integral. In this paper, we extend it to a generalized Stieltjes integral dg of a point function f with respect to a function g of divisions of an interval. Then we prove for this integral the standard results in the theory of integration, including the controlled convergence theorem.      

一类推广的KdV方程的新行波解

本文研究了一类推广的Kd V方程的行波解求解的问题.利用新的G展开法,并借助Mathematica计算软件,获得了该方程的含有多个任意参数的新的行波解,分别为三角函数解、双曲函数解、有理函数解和指数函数解,扩大了该类方程的解的范围.     

Tail index estimation for heavy tails: accommodation of bias in the excesses over a high threshold

In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called “maximum likelihood” estimators of the tail index γ. In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar “maximum likelihood” estimator of a positive tail index γ, based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study. Research partially supported by FCT/POCTI and POCI/FEDER.     

The Dirichlet problem for Baire-one functions

Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.     

ON A NEW SINGULAR DIRECTION OF QUASIMEROMORPHIC MAPPINGS

By applying Ahlfors＇ theory of covering surface, we establish a fundamental inequality for quasimeromorphic mapping in an angular domain. As an application, we prove the existence of a new singular direction for quasimeromorphic mapping f, namely, a precise S direction, for which the spherical characteristic function S（r, f） is used as a comparison function.     

The Maximum Lq-Likelihood Method: An Application to Extreme Quantile Estimation in Finance

Estimating financial risk is a critical issue for banks and insurance companies. Recently, quantile estimation based on extreme value theory (EVT) has found a successful domain of application in such a context, outperforming other methods. Given a parametric model provided by EVT, a natural approach is maximum likelihood estimation. Although the resulting estimator is asymptotically efficient, often the number of observations available to estimate the parameters of the EVT models is too small to make the large sample property trustworthy. In this paper, we study a new estimator of the parameters, the maximum Lq-likelihood estimator (MLqE), introduced by Ferrari and Yang (Estimation of tail probability via the maximum Lq-likelihood method, Technical Report 659, School of Statistics, University of Minnesota, 2007 ). We show that the MLqE outperforms the standard MLE, when estimating tail probabilities and quantiles of the generalized extreme value (GEV) and the generalized Pareto (GP) distributions. First, we assess the relative efficiency between the MLqE and the MLE for various sample sizes, using Monte Carlo simulations. Second, we analyze the performance of the MLqE for extreme quantile estimation using real-world financial data. The MLqE is characterized by a distortion parameter q and extends the traditional log-likelihood maximization procedure. When q→1, the new estimator approaches the traditional maximum likelihood estimator (MLE), recovering its desirable asymptotic properties; when q ≠ 1 and the sample size is moderate or small, the MLqE successfully trades bias for variance, resulting in an overall gain in terms of accuracy (mean squared error).      

A new approximate functional equation for Hurwitz zeta function for rational parameter

For Hurwitz zeta function ζ(s, (a/k)) witha = 1,2,3,…,k, we obtain a new simple approximate functional equation (uniform ink andt) in critical strip. Our method should prove to be an alternative approach to Atkinson’s method in dealing with , whereL(s, x) is Dirichlet L-series moduloq and s = σ +it.     

Some comments on a paper onk-HNBUE life distributions

Summary In a recent paper by A. P. Basu and N. Ebrahimi (1984, Ann. Inst. Statist. Math., A, 36, 87–100) a new class of life distributions calledk-HNBUE (with dualk-HNWUE) is introduced. Closure properties and bounds on the moments and on the survival function to ak-HNBUE (k-HNWUE) life distribution are presented. However, some of the results presented are incorrect. This research was supported by Swedish Natural Science Research Council Post Doctoral Fellowship F-PD 1564-100.