Statistical Analysis of a Spatial Counting Process Modelling Crystallization of Polymers

The nucleation phase of the crystallization of polymers is described in terms of a stochastic spatial counting process, whose intensity depends upon the available volume. Estimation of the relevant parameters of the process are obtained via the maximum likelihood method [6]. The asymptotic properties of the estimators (proved in [6]) are applied to study their qualitative behaviour, as a function of the available volume and time. In this paper, a goodness of fit of the stochastic model proposed has been carried out via a Kolmogorov-Smirnov approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimisation of linear unbiased intensity estimators for point processes

A general non-stationary point process whose intensity function is given up to unknown numerical factor λ is considered. As an alternative to the conventional estimator of λ based on counting the points, we consider general linear unbiased estimators of λ given by sums of weights associated with individual points. A necessary and sufficient condition for a linear, unbiased estimator for the intensity λ to have the minimum variance is determined. It is shown that there are “nearly” no other processes than Poisson and Cox for which the unweighted estimator of λ, which counts the points only, is optimal. The properties of the optimal estimator are illustrated by simulations for the Matérn cluster and the Matérn hard-core processes. This research was partially supported by Grant Agency of Czech Republic, project No. 201/03/D062.     

A Robust Heuristic Estimator for the Period of a Poisson Intensity Function

While there are a number of methods for estimating the intensity of a cyclic point process, these assume prior estimation of the period itself. The standard method for the latter is the periodogram, or spectral analysis, approach. This is a parametric method which is sensitive to the form, in particular the number of peaks per cycle, of the intensity. We construct a family of nonparametric estimators for the period of a cyclic Poisson process, with the object of robustness against the form of the intensity. These are tested, along with the standard periodogram estimate and an earlier nonparametric estimator, on simulated data from a range of intensity functions. While the nonparametric estimators presently lack the well-developed asymptotic and statistical properties of the periodogram, the best of them is almost as accurate as the periodogram for the unimodal intensity cycles on which the latter is based. Whereas the periodogram cannot handle multimodal cycles at all, the better nonparametric estimators are reasonably accurate, and sometimes err by estimating multiples of the period rather than divisors, errors that are arguably less damaging. We conclude with some remarks concerning the derivation of asymptotic properties for our nonparametric estimator.     

Maximum likelihood estimation in a partially observed stratified regression model with censored data

The stratified proportional intensity model generalizes Cox’s proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also obtained.     

Proportional hazards regression under progressive Type-II censoring

This paper proposes an inferential method for the semiparametric proportional hazards model for progressively Type-II censored data. We establish martingale properties of counting processes based on progressively Type-II censored data that allow to derive the asymptotic behavior of estimators of the regression parameter, the conditional cumulative hazard rate functions, and the conditional reliability functions. A Monte Carlo study and an example are provided to illustrate the behavior of our estimators and to compare progressive Type-II censoring sampling plans with classical Type-II right censoring sampling plan.     

On information inequalities in sequential estimation for stochastic processes

Information inequalities in a general sequential model for stochastic processes are presented by applying the approach to estimation through estimating functions. Using this approach, Bayesian versions of the information inequalities are also obtained. In particular, exponential-family processes and counting processes are considered. The results are useful to find optimum properties of parameter estimators. The assertions are of great importance for describing estimators in failure-repair models in both Bayes approach and the nuisance parameter case.     

Wavelets-based estimation of nonlinear canonical analysis

Using a wavelets-based estimator of the bivariate density, we introduce an estimation method for nonlinear canonical analysis. Consistency of the resulting estimators of the canonical coefficients and the canonical functions is established. Under some conditions, asymptotic normality results for these estimators are obtained. Then it is shown how to compute in practice these estimators by usingmatrix computations, and the finite-sample performance of the proposed method is evaluated through simulations.     

On properties of estimators in nonregular situations for Poisson processes

We consider the problem of parameter estimation by observations of an inhomogeneous Poisson process. It is well known that if regularity conditions are fulfilled, then the maximum likelihood and Bayesian estimators are consistent, asymptotically normal, and asymptotically efficient. These regularity conditions can be roughly presented as follows: (a) the intensity function of the observed process belongs to a known parametric family of functions, (b) the model is identifiable, (c) the Fisher information is a positive continuous function, (d) the intensity function is sufficiently smooth with respect to the unknown parameter, and (e) this parameter is an interior point of the interval. We are interested in properties of estimators for which these regularity conditions are not fulfilled. More precisely, we present a review of results which correspond to the rejection of these conditions one by one and show how properties of the MLE and Bayesian estimators change. The proofs of these results are essentially based on some general results by Ibragimov and Khasminskii. Bibliography: 9 titles.     

Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities

 In this paper, we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous Poisson process. We study consequently the adaptive properties of penalized projection estimators. At first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants. The crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of Poisson processes which are analogous to Talagrand's inequalities for empirical processes. Received: 24 April 2001 / Revised version: 9 October 2002 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60E15, 62G05, 62G07 Key words or phrases: Inhomogeneous Poisson process – Concentration inequalities – Model selection – Penalized projection estimator – Adaptive estimation     

Statistical estimation for some dividend problems under the compound Poisson risk model

In this paper, we consider some dividend problems in the classical compound Poisson risk model under a constant barrier dividend strategy. Suppose that the Poisson intensity for the claim number process and the distribution for the individual claim sizes are both unknown. We use the COS method to study the statistical estimation for the expected present value of dividend payments before ruin and the expected discounted penalty function. The convergence rates under large sample setting are derived. Some simulation results are also given to show effectiveness of the estimators under finite sample setting.     

基于纵向数据的半参数变系数部分线性回归模型

纵向数据是数理统计研究中的复杂数据类型之一0,在生物、医学和经济学中具有广泛的应用．在实际中经常需要对纵向数据进行统计分析和建模．文章讨论了纵向数据下的半参数变系数部分线性回归模型,这里的纵向数据的在纵向观察在时间上可以是不均等的,也可看成是按某一随机过程来发生．所研究的半参数变系数模型包括了许多半参数模型,比如部分线性模型和变系数模型等．利用计数过程理论和局部线性回归方法,对于纵向数据下半参数变系数进行了统计推断,给出了参数分量和非参数分量的profile最小二乘估计,研究了这些估计的渐近性质,获得这些估计的相合性和渐近正态性．     

固定设计下半参函数关系模型参数估计的相合性

对于固定设计下的半参数函数关系模型,利用广义最小二乘法和一般的非参数权估计方法,得出了未知参数和未知函数的估计.在一定条件下,证明了它们的强相合性及其p(≥2)阶平均相合性.     

SUR模型回归系数的估计

本文证明了一个关于SUR模型回归系数最小方差线性无偏估计(MVLUE)的充要条件,并利用此充要条件讨论了几类SUR模型回归系数的MVLUE估计及两步估计.在方法上避免了与分块矩阵求逆有关的运算,所得结论推广和完善了已有的一些结果.     

截断样本下的强率函数和密度函数核估计的渐进正态性

本文用[1]发展的计数过程去研究截断样本下强率函数核估计的渐进正态性．在弱于[7]和[10]的条件下，得到了更一般的结果．接着我们将这种方法运用到密度函数核估计，在较弱的条件下，得到了截断样本下密度函数核估计的渐进正态性．     

ψ-混合样本下含附加信息时条件分位数估计的渐近性质

本文利用经验似然方法构造了含附加信息时条件分位数的一类估计,并证明了估计的渐近正态性且渐近方差不大于通常核估计的渐近方差.     

NA样本下半参数EV模型小波估计的渐近性简

In this paper, we research the semiparametric EV model under NA samples. Some estimators of the parameter, nonparameter and the variance function are established by the wavelet smoothing method. Under some general conditions, the strong consistency and the asymptotic normality of wavelet estimators are studied.     

Local asymptotic mixed normality for semimartingale experiments

Summary We give conditions for local asymptotic mixed normality of experiments when the observed process is a semimartingale and the observation time increases to infinity. As a consequence we obtain asymptotic efficiency of various estimators. Several special models for counting process,s, diffusion processes and diffusions with jumps are studied.Research supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft     

One-step jackknife for M-estimators computed using Newton's method

To estimate the dispersion of an M-estimator computed using Newton's iterative method, the jackknife method usually requires to repeat the iterative process n times, where n is the sample size. To simplify the computation, one-step jackknife estimators, which require no iteration, are proposed in this paper. Asymptotic properties of the one-step jackknife estimators are obtained under some regularity conditions in the i.i.d. case and in a linear or nonlinear model. All the one-step jackknife estimators are shown to be asymptotically equivalent and they are also asymptotically equivalent to the original jackknife estimator. Hence one may use a dispersion estimator whose computation is the simplest. Finite sample properties of several one-step jackknife estimators are examined in a simulation study.The research was supported by Natural Sciences and Engineering Research Council of Canada.     

On stochastic estimation

We consider a local random searching method to approximate a root of a specified equation. If such roots, which can be regarded as estimators for the Euclidean parameter of a statistical experiment, have some asymptotic optimality properties, the local random searching method leads to asymptotically optimal estimators in such cases. Application to simple first order autoregressive processes and some simulation results for such models are also included.     

固定设计的半参数函数关系模型

本文在固定设计下研究半参数变量含误差模型，利用权函数和加权最小二乘法得出未知参数的估计，在一定的条件下证明了估计的强相合性。