Asymptotic distribution of martingale estimators for a class of epidemic models

This article is a contribution to the asymptotic inference on the parameters of a quite general class of stochastic models for the spread of epidemics developing in closed populations. Various epidemic models are contained within our framework, for instance, a stochastic version of the Kermack and McKendrick model and the SIS epidemic model. Each model belonging to this class, which consists in a family of discrete-time stochastic process, contains certain parameters to be estimated by means of martingale estimators. Some particular cases defined by means of Markov chains are included in our setting. The main aim of this work is to prove consistency and asymptotic normality of these estimators. Some hypothesis tests based on the main results are also shown.     

Statistical Inference on a Stochastic Epidemic Model

In this work, we develop statistical inference for the parameters of a discrete-time stochastic SIR epidemic model. We use a Markov chain for describing the dynamic behavior of the epidemic. Specifically, we propose estimators for the contact and removal rates based on the maximum likelihood and martingale methods, and establish their asymptotic distributions. The obtained results are applied in the statistical analysis of the basic reproduction number, a quantity that is useful in establishing vaccination policies. In order to evaluate the population size for which the results are useful, a numerical study is carried out. Finally, a comparison of the maximum likelihood and martingale estimators is conducted by means of Monte Carlo simulations.     

Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

This article deals with quasi- and pseudo-likelihood estimation for a class of continuous-time multi-type Markov branching processes observed at discrete points in time. “Conventional” and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples.     

Asymptotic variance approximations for invariant estimators in uncertain asset-pricing models

This article derives explicit expressions for the asymptotic variances of the maximum likelihood and continuously-updated GMM estimators in models that may not satisfy the fundamental asset-pricing restrictions in population. The proposed misspecification-robust variance estimators allow the researcher to conduct valid inference on the model parameters even when the model is rejected by the data. While the results for the maximum likelihood estimator are only applicable to linear asset-pricing models, the asymptotic distribution of the continuously-updated GMM estimator is derived for general, possibly nonlinear, models. The large corrections in the asymptotic variances, that arise from explicitly incorporating model misspecification in the analysis, are illustrated using simulations and an empirical application.     

Sampling Adjusted Analysis of Dynamic Additive Regression Models for Longitudinal Data

We consider a modelling approach to longitudinal data that aims at estimating flexible covariate effects in a model where the sampling probabilities are modelled explicitly. The joint modelling yields simple estimators that are easy to compute and analyse, even if the sampling of the longitudinal responses interacts with the response level. An incorrect model for the sampling probabilities results in biased estimates. Non-representative sampling occurs, for example, if patients with an extreme development (based on extreme values of the response) are called in for additional examinations and measurements. We allow covariate effects to be time-varying or time-constant. Estimates of covariate effects are obtained by solving martingale equations locally for the cumulative regression functions. Using Aalen's additive model for the sampling probabilities, we obtain simple expressions for the estimators and their asymptotic variances. The asymptotic distributions for the estimators of the non-parametric components as well as the parametric components of the model are derived drawing on general martingale results. Two applications are presented. We consider the growth of cystic fibrosis patients and the prothrombin index for liver cirrhosis patients. The conclusion about the growth of the cystic fibrosis patients is not altered when adjusting for a possible non-representativeness in the sampling, whereas we reach substantively different conclusions about the treatment effect for the liver cirrhosis patients.     

Estimation in multitype epidemics

A multitype epidemic model is analysed assuming proportionate mixing between types. Estimation procedures for the susceptibilities and infectivities are derived for three sets of data: complete data, meaning that the whole epidemic process is observed continuously; the removal processes are observed continuously; only the final state is observed. Under the assumption of a major outbreak in a population of size n it is shown that, for all three data sets, the susceptibility estimators are always efficient, i.e. consistent with a √ n rate of convergence. The infectivity estimators are 'in most cases' respectively efficient, efficient and unidentifiable. However, if some susceptibilities are equal then the corresponding infectivity estimators are respectively barely consistent (√log( n ) rate of convergence), not consistent and unidentifiable. The estimators are applied to simulated data.     

Parameter estimation in regression for long-term survival rate from censored data

In recent years, regression models have been shown to be useful for predicting the long-term survival probabilities of patients in clinical trials. The importance of a regression model is that once the regression parameters are estimated information about the regressed quantity is immediate. A simple estimator is proposed for the regression parameters in a model for the long-term survival rate. The proposed estimator is seen to arise from an estimating function that has the missing information principle underlying its construction. When the covariate takes values in a finite set, the proposed estimating function is equivalent to an ad hoc estimating function proposed in the literature. However, in general, the two estimating functions lead to different estimators of the regression parameter. For discrete covariates, the asymptotic covariance matrix of the proposed estimator is simple to calculate using standard techniques involving the predictable covariation process of martingale transforms. An ad hoc extension to the case of a one-dimensional continuous covariate is proposed. Simplicity and generalizability are two attractive features of the proposed approach. The last mentioned feature is not enjoyed by the other estimator.     

A Test to Detect Within-family Infectivity when the Whole Epidemic Process is Observed

An epidemic model for the spread of an infectious disease in a population of families is considered. The score test of the hypothesis that there is no higher infectivity between family members is constructed under the assumption that the epidemic process is observed continuously up to some time t . The score process is a martingale as a function of t and by letting the number of families tend to infinity, a central limit theorem for the process can be proved. The central limit theorem not only justifies a normal approximation of the test statistic—it also suggests a smaller variance estimator than expected.     

Semiparametric regression for restricted mean residual life under right censoring

A mean residual life function (MRLF) is the remaining life expectancy of a subject who has survived to a certain time point. In the presence of covariates, regression models are needed to study the association between the MRLFs and covariates. If the survival time tends to be too long or the tail is not observed, the restricted mean residual life must be considered. In this paper, we propose the proportional restricted mean residual life model for fitting survival data under right censoring. For inference on the model parameters, martingale estimating equations are developed, and the asymptotic properties of the proposed estimators are established. In addition, a class of goodness-of-fit test is presented to assess the adequacy of the model. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and the approach is applied to a set of real life data collected from a randomized clinical trial.     

Analysis of transformation models with right-truncated data

In many applications, statistical data are frequently observed subject to a retrospective sampling criterion resulting in right-truncated data. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of right-truncated data. We proposed two estimators for regression coefficients. The first estimator is based on martingale estimating equations. The second estimator is based on the conditional likelihood function given the truncation times. The asymptotic properties of both estimators are derived. The finite sample performance is examined through a simulation study.     

A BETA-BINOMIAL MODEL FOR ESTIMATING THE SIZE OF A HETEROGENEOUS POPULATION

This paper compares the properties of various estimators for a beta‐binomial model for estimating the size of a heterogeneous population. It is found that maximum likelihood and conditional maximum likelihood estimators perform well for a large population with a large capture proportion. The jackknife and the sample coverage estimators are biased for low capture probabilities. The performance of the martingale estimator is satisfactory, but it requires full capture histories. The Gibbs sampler and Metropolis‐Hastings algorithm provide reasonable posterior estimates for informative priors.     

The exponential–Weibull lifetime distribution

In this paper, we propose a new three-parameter model called the exponential–Weibull distribution, which includes as special models some widely known lifetime distributions. Some mathematical properties of the proposed distribution are investigated. We derive four explicit expressions for the generalized ordinary moments and a general formula for the incomplete moments based on infinite sums of Meijer's G functions. We also obtain explicit expressions for the generating function and mean deviations. We estimate the model parameters by maximum likelihood and determine the observed information matrix. Some simulations are run to assess the performance of the maximum likelihood estimators. The flexibility of the new distribution is illustrated by means of an application to real data.     

Strong consistency of conditional least squares estimators in multiple regime threshold autoregressive models

In this paper we analyse the conditional least squares estimators of the parameters of a multiple regime threshold AR(1) model and prove that under certain conditions these are strongly consistent. We assume that the error process in each regime is amartingale difference sequence. Then we deal with strong consistency of the natural estimator of the error variance in each regime.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

Estimation of the area under ROC curve with censored data

The area under the receiver operating characteristic curve is the most commonly used summary measure of diagnostic accuracy for a continuous-scale diagnostic test. In this paper, we develop methods to estimate the area under the curve (AUC) with censored data. Based on two different integration representations of this parameter, two nonparametric estimators are defined by the “plug in” method. Both the proposed estimators are shown to be asymptotically normal based on counting process and martingale theory. A simulation study is conducted to evaluate the performances of the proposed estimators.     

Bias Reduction for the Maximum Likelihood Estimator of the Parameters of the Generalized Rayleigh Family of Distributions

We derive analytic expressions for the biases, to O(n^{? 1}), of the maximum likelihood estimators of the parameters of the generalized Rayleigh distribution family. Using these expressions to bias-correct the estimators is found to be extremely effective in terms of bias reduction, and generally results in a small reduction in relative mean squared error. In general, the analytic bias-corrected estimators are also found to be superior to the alternative of bias-correction via the bootstrap.     

An evaluation by simulation of alternative estimators for the marginal Fisher distribution

The Fisher distribution is a standard model for directional data (or spherical data). In some cases though, only the co-latitudes can be observed, resulting in a sample of observations from the corresponding marginal distribution. This paper reports on an extensive simulation to compare and evaluate the robustness of 11 test-statistics corresponding to various estimators of the parameters of this distribution. The estimators include Maximum Likelihood and Moment-type estimators, as well as sample means and variances based on approximations to the marginal Fisher distribution. Of the test-statistics considered, the Likelihood-Ratio statistic was the only one whose sampling distribution remained close to its asymptotic distribution for all parameter values and sample sizes considered. In general, the other statistics were close to their approximate distributions only when ksin2?0, was fairly large. The paper includes details on the computational methods for finding the Maximum Likelihood and Moment estimators, and concludes with some practical advice on the choice of estimation procedure.     

Simplified Estimating Functions for Diffusion Models with a High-dimensional Parameter

We consider estimating functions for discretely observed diffusion processes of the following type: for one part of the parameter of interest we propose to use a simple and explicit estimating function of the type studied by Kessler (2000); for the remaining part of the parameter we use a martingale estimating function. Such an approach is particularly useful in practical applications when the parameter is high-dimensional. It is also often necessary to supplement a simple estimating function by another type of estimating function because only the part of the parameter on which the invariant measure depends can be estimated by a simple estimating function. Under regularity conditions the resulting estimators are consistent and asymptotically normal. Several examples are considered in order to demonstrate the idea of the estimating procedure. The method is applied to two data sets comprising wind velocities and stock prices. In one example we also propose a general method for constructing diffusion models with a prescribed marginal distribution which have a flexible dependence structure.     

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.     

Estimation of location extremes within general families of scale mixtures

In this paper, we study the estimation of the minimum and maximum location parameters, respectively, representing the minimum guaranteed lifetime of series and parallel systems of components, within a general class of scale mixtures. The conditional or underlying distribution has only the primary restriction of being a location-scale family with positive support. The mixing distribution is also quite general in that we only assume that it has positive support and finite second moment. For demonstrative purposes several special cases are highlighted such as the gamma, inverse-Gaussian, and discrete mixture. Various estimators, including bootstrap bias corrected estimators, are compared with respect to both mean-squared-error and Pitman's measure of closeness.