Nonparametric Inference for a Partially Observed Compartmental Process

Methods of nonparametric inference are proposed for a process with two transient and three absorbing states. It is assumed that the time of transitions between the transient states are unobservable. One area of applications is in epidemiology where the transient states correspond to healthy and ill, while the absorbing states correspond to types of death. It is the onset of illness which is not observable. An estimate is given for a cumulative hazard rate between the transient states, the exit hazard rates are estimated at a specific point in time and a statistic for comparing exit rates from the transient states is given.     

Non-parametric Estimation of a Survival Function with Two-stage Design Studies

Abstract.  The two-stage design is popular in epidemiology studies and clinical trials due to its cost effectiveness. Typically, the first stage sample contains cheaper and possibly biased information, while the second stage validation sample consists of a subset of subjects with accurate and complete information. In this paper, we study estimation of a survival function with right-censored survival data from a two-stage design. A non-parametric estimator is derived by combining data from both stages. We also study its large sample properties and derive pointwise and simultaneous confidence intervals for the survival function. The proposed estimator effectively reduces the variance and finite-sample bias of the Kaplan–Meier estimator solely based on the second stage validation sample. Finally, we apply our method to a real data set from a medical device postmarketing surveillance study.     

A Partial Likelihood Estimator of Vaccine Efficacy

A partial likelihood method is proposed for estimating vaccine efficacy for a general epidemic model. In contrast to the maximum likelihood estimator (MLE) which requires complete observation of the epidemic, the suggested method only requires information on the sequence in which individuals are infected and not the exact infection times. A simulation study shows that the method performs almost as well as the MLE. The method is applied to data on the infectious disease mumps.     

Upper bounds for ruin probabilities under model uncertainty

Abstract

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.     

Testing recapture model MT for departure from homogeneity

This paper presents a method of testing for animal heterogeneity when the capture effort is not constant from occasion to occasion. The basic set of statistics used is a lower triangular matrix whose (I,k)th element is the proportion of those animals caught on the next (i+1)the occasion who have been caught k times until now. Both theoretical and simulation evidence is presented that the new procedure is superior to methods previously suggested. The method may breakdown for certain kinds of behavioural response.     

On the Local Polynomial Estimators of the Counting Process Intensity Function and its Derivatives

Abstract. We consider the properties of the local polynomial estimators of a counting process intensity function and its derivatives. By expressing the local polynomial estimators in a kernel smoothing form via effective kernels, we show that the bias and variance of the estimators at boundary points are of the same magnitude as at interior points and therefore the local polynomial estimators in the context of intensity estimation also enjoy the automatic boundary correction property as they do in other contexts such as regression. The asymptotically optimal bandwidths and optimal kernel functions are obtained through the asymptotic expressions of the mean square error of the estimators. For practical purpose, we suggest an effective and easy‐to‐calculate data‐driven bandwidth selector. Simulation studies are carried out to assess the performance of the local polynomial estimators and the proposed bandwidth selector. The estimators and the bandwidth selector are applied to estimate the rate of aftershocks of the Sichuan earthquake and the rate of the Personal Emergency Link calls in Hong Kong.     

HETEROGENEITY AMONG SUSCEPTIBLES IN EPIDEMIC MODELS

When the infection rate associated with an epidemic appears to decline over time, one explanation is a constant level of infectiousness combined with heterogeneity among the susceptible population. In this paper we consider random effects models for such heterogeneity, particularly in discrete time. Maximum likelihood techniques are discussed as well as a more convenient approach based on martingale estimating equations. An application to data on a smallpox outbreak is considered.     

On almost sure convergence for sums of stochastic sequence

In this work we discuss almost sure convergence for sums of arbitrarily dependent stochastic sequence under different conditions of Chung’s type. Our approach is based on the stopping time technique and the theorem of convergence for martingale difference sequence. Meanwhile, the results here include some relevant classical conclusions.     

Checking the Short‐Term and Long‐Term Hazard Ratio Model for Survival Data

Abstract. The short‐term and long‐term hazard ratio model includes the proportional hazards model and the proportional odds model as submodels, and allows a wider range of hazard ratio patterns compared with some of the more traditional models. We propose two omnibus tests for checking this model, based, respectively, on the martingale residuals and the contrast between the non‐parametric and model‐based estimators of the survival function. These tests are shown to be consistent against any departure from the model. The empirical behaviours of the tests are studied in simulations, and the tests are illustrated with some real data examples.