A List Sequential Sampling Method Suitable for Real-Time Sampling

Abstract.  A flexible list sequential π ps sampling method is introduced and studied. It can reproduce any given sampling design without replacement, of fixed or random sample size. The method is a splitting method and uses successive updating of inclusion probabilities. The main advantage of the method is in real-time sampling situations where it can be used as a powerful alternative to Bernoulli and Poisson sampling and can give any desired second-order inclusion probabilities and thus considerably reduce the variability of the sample size.     

Estimating the Basic Reproductive Number in the General Epidemic Model with an Unknown Initial Number of Susceptible Individuals

Abstract. In any epidemic, there may exist an unidentified subpopulation which might be naturally immune or isolated and who will not be involved in the transmission of the disease. Estimation of key parameters, for example, the basic reproductive number, without accounting for this possibility would underestimate the severity of the epidemics. Here, we propose a procedure to estimate the basic reproductive number ( R ₀ ) in an epidemic model with an unknown initial number of susceptibles. The infection process is usually not completely observed, but is reconstructed by a kernel‐smoothing method under a counting process framework. Simulation is used to evaluate the performance of the estimators for major epidemics. We illustrate the procedure using the Abakaliki smallpox data.     

Determination of the functional form of the relationship of covariates to the log hazard ratio in a Cox model

In this paper, we review available methods for determination of the functional form of the relation between a covariate and the log hazard ratio for a Cox model. We pay special attention to the detection of influential observations to the extent that they influence the estimated functional form of the relation between a covariate and the log hazard ratio. Our paper is motivated by a data set from a cohort study of lung cancer and silica exposure, where the nonlinear shape of the estimated log hazard ratio for silica exposure plotted against cumulative exposure and hereafter referred to as the exposure–response curve was greatly affected by whether or not two individuals with the highest exposures were included in the analysis. Formal influence diagnostics did not identify these two individuals but did identify the three highest exposed cases. Removal of these three cases resulted in a biologically plausible exposure–response curve.     

Fully Nonparametric Estimation of Scalar Diffusion Models

We propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process. We show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter. The asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens. We prove almost sure consistency and weak convergence to mixtures of normal laws, where the mixing variates depend on the chronological local time of the underlying diffusion process, that is the random time spent by the process in the vicinity of a generic spatial point. The estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes.     

On the complete convergence for martingale difference sequence

In this paper, we establish several generalized results on complete convergence for martingale difference sequence, which include some well-known results.     

带干扰的双险种Cox风险模型

给出了一个较一般的风险模型即带干扰的双险种Cox风险模型,并运用鞅的方法得出了保险公司的最终破产概率ψ(u)的不等式,使得用它来研究保险公司的盈利更符合实际情况,更具有实际意义.     

Estimating the Nitrous Oxide Emission Rate from the Soil Surface by Means of a Diffusion Model

Soil cover methods are probably the most widely used methods for measuring the nitrous oxide emission rate from the soil surface. The methodology involves estimation of the emission rate from repeated measurements of the nitrous oxide concentration beneath a soil cover. Based on a deterministic model proposed by Hutchinson & Mosier (1981) we propose to use a diffusion process as a stochastic model for the evolution of the nitrous oxide concentrations beneath a soil cover. From this model we derive methods for statistical inference about the emission rate that significantly extend the method proposed by Hutchinson & Mosier (1981). In particular, the derived methods provide solutions to important problems with the method proposed by Hutchinson & Mosier (1981).     

Discretely Observed Diffusions: Classes of Estimating Functions and Small Δ-optimality

Ergodic diffusions in several dimensions, depending on an unknown multivariate parameter are considered. For estimation, when the diffusion is observed only at finitely many equidistant time points, unbiased estimating functions leading to consistent and asymptotically Gaussian estimators are used. Different types of estimating functions are discussed and the concept of small Δ-optimality is introduced to help select good estimating functions. Explicit criteria for small Δ-optimality are given. Also some exact optimality conditions are presented as well as, for one-dimensional diffusions, methods for improving estimators using time reversibility.     

Asymptotic properties for estimates of nonparametric regression model with martingale difference errors

Consider the nonparametric regression model with martingale difference errors. Nonparametric estimator g _n (x) of regression function g(x) will be introduced, and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g _n (x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors.     

A goodness-of-fit test for the stratified proportional hazards model for survival data

Abstract

Goodness-of-fit testing is addressed in the stratified proportional hazards model for survival data. A test statistic based on within-strata cumulative sums of martingale residuals over covariates is proposed and its asymptotic distribution is derived under the null hypothesis of model adequacy. A Monte Carlo procedure is proposed to approximate the critical value of the test. Simulation studies are conducted to examine finite-sample performance of the proposed statistic.