Logistic regression in meta-analysis using aggregate data

We derived two methods to estimate the logistic regression coefficients in a meta-analysis when only the 'aggregate' data (mean values) from each study are available. The estimators we proposed are the discriminant function estimator and the reverse Taylor series approximation. These two methods of estimation gave similar estimators using an example of individual data. However, when aggregate data were used, the discriminant function estimators were quite different from the other two estimators. A simulation study was then performed to evaluate the performance of these two estimators as well as the estimator obtained from the model that simply uses the aggregate data in a logistic regression model. The simulation study showed that all three estimators are biased. The bias increases as the variance of the covariate increases. The distribution type of the covariates also affects the bias. In general, the estimator from the logistic regression using the aggregate data has less bias and better coverage probabilities than the other two estimators. We concluded that analysts should be cautious in using aggregate data to estimate the parameters of the logistic regression model for the underlying individual data.     

Reweighting approximate GM estimators: Asymptotics and residual-based graphics

The iterative weighted least squares algorithm is handy for solving generalized estimating equations. In some situations it may be desirable to limit the number of iterations to a fixed finite number, for instance, to keep the breakdown point under control. Such a scheme is called reweighting. Usually reweighting leads to a different large sample theory than full iteration, and the reweighted estimator may inherit deficiencies of the starting value. When might the reweighting scheme work? To answer this question we define a broad class of estimators, namely, approximate GM estimators, and we show that reweighting leads to the same large sample theory as full iteration within this class. As an example, we provide conditions under which one-step Newton-Raphson estimators are approximate GM estimators. We then use the reweighting to construct residual-based graphics for approximate GM estimates, adapting weighted residual plots that have been proposed previously, and developing new plots to provide complementary views of the data.     

Parameter Estimation in Conditional Heteroscedastic Models

Abstract

We study asymptotics of parameter estimates in conditional heteroscedastic models. The estimators considered are those obtained by minimizing certain functionals and those obtained by solving estimation equations. We establish consistency and derive asymptotic limit laws of the estimators. Condition under which the limit law is normal is studied. Further, bootstrap for these estimators is discussed. The limiting distribution of the estimators is not necessary always normal, and we present a real data example to illustrate this.     

ϵ-Admissible Estimators for Normal and Poisson Means

The criterion of admissibility has been considered as one of the most important criterion in decision theory and many important results have been contributed in this direction. In this article, we propose a more flexible criterion, so-called ?-admissibility (which can be considered as weak admissibility), which generates a monotone sequence of classes of estimators. The limit of this sequence, class of 0⁺-admissible estimators, is the smallest class including the class of usual admissible estimators, which also belongs to the monotone sequence. Some sufficient and necessary conditions are proposed for ?-admissibility and 0⁺-admissibility. Under some weighted square loss, it can be shown that the usual MLE is 0⁺-admissible for the multivariate normal distribution and the multivariate Poisson distribution.     

Bayes methods in the ecological fallacy cntext:estimation of individual correlation from aggregate data

The ecological fallacy is related to Simpson's paradox (1951) where relationships among group means may be counterintuitive and substantially different from relationships within groups, where the groups are usually geographic entities such as census tracts. We consider the problem of estimating the correlation between two jointly normal random variables where only ecological data (group means) are available. Two empirical Bayes estimators and one fully Bayesian estimator are derived and compared with the usual ecological estimator, which is simply the Pearson correlation coefficient of the group sample means. We simulate the bias and mean squared error performance of these estimators, and also give an example employing a dataset where the individual level data are available for model checking. The results indicate superiority of the empirical Bayes estimators in a variety of practical situations where, though we lack individual level data, other relevant prior information is available.     

Delay Estimation for Some Stationary Diffusion-type Processes

In this paper the asymptotic behaviour of the maximum likelihood and Bayesian estimators of a delay parameter is studied. The observed process is supposed to be the solution of a linear stochastic differential equation with one time delay term. It is shown that these estimators are consistent and their limit distributions are described. The behaviour of the estimators is similar to the behaviour of corresponding estimators in change-point problems. The question of asymptotical efficiency is also discussed.     

A semiparametric additive rate model for a modulated renewal process

Recurrent event data from a long single realization are widely encountered in point process applications. Modeling and analyzing such data are different from those for independent and identical short sequences, and the development of statistical methods requires careful consideration of the underlying dependence structure of the long single sequence. In this paper, we propose a semiparametric additive rate model for a modulated renewal process, and develop an estimating equation approach for the model parameters. The asymptotic properties of the resulting estimators are established by applying the limit theory for stationary mixing sequences. A block-based bootstrap procedure is presented for the variance estimation. Simulation studies are conducted to assess the finite-sample performance of the proposed estimators. An application to a data set from a cardiovascular mortality study is provided.     

Dual divergences estimation for censored survival data

This paper is devoted to robust estimation based on dual divergences estimators for parametric models in the framework of right censored data. We give limit laws of the proposed estimators and examine their asymptotic properties through a simulation study.     

Bias reduction in the estimation of a shape second-order parameter of a heavy-tailed model

In extreme value theory, the shape second-order parameter is a quite relevant parameter related to the speed of convergence of maximum values, linearly normalized, towards its limit law. The adequate estimation of this parameter is vital for improving the estimation of the extreme value index, the primary parameter in statistics of extremes. In this article, we consider a recent class of semi-parametric estimators of the shape second-order parameter for heavy right-tailed models. These estimators, based on the largest order statistics, depend on a real tuning parameter, which makes them highly flexible and possibly unbiased for several underlying models. In this article, we are interested in the adaptive choice of such tuning parameter and the number of top order statistics used in the estimation procedure. The performance of the methodology for the adaptive choice of parameters is evaluated through a Monte Carlo simulation study.     

Asymptotic results for random coefficient bifurcating autoregressive processes

The purpose of this paper is to study the asymptotic behaviour of the weighted least-squares estimators of the unknown parameters of random coefficient bifurcating autoregressive processes. Under suitable assumptions on the immigration and the inheritance, we establish the almost sure convergence of our estimators, as well as a quadratic strong law and central limit theorems. Our study mostly relies on limit theorems for vector-valued martingales.     

Nonparametric analysis of aggregate loss models

This paper describes a nonparametric approach to make inferences for aggregate loss models in the insurance framework. We assume that an insurance company provides a historical sample of claims given by claim occurrence times and claim sizes. Furthermore, information may be incomplete as claims may be censored and/or truncated. In this context, the main goal of this work consists of fitting a probability model for the total amount that will be paid on all claims during a fixed future time period. In order to solve this prediction problem, we propose a new methodology based on nonparametric estimators for the density functions with censored and truncated data, the use of Monte Carlo simulation methods and bootstrap resampling. The developed methodology is useful to compare alternative pricing strategies in different insurance decision problems. The proposed procedure is illustrated with a real dataset provided by the insurance department of an international commercial company.     

Nonparametric estimation of a survival function under progressive Type-I multistage censoring

Failure time data subject to three progressive Type-I multistage censoring schemes are studied. Product limit estimators are proposed for the estimation of the survival function. It is shown that the resulting estimators are asymptotically equivalent to the corresponding estimators where the data are subject to a random iid right censoring scheme. Many well-known results on confidence bands and tests for randomly right censored data hold for these progressive censoring schemes.     

Functional version of the delta method and its application

Nonparametric estimation of the time varying parameter of a semimartingal e model is considered. Stochastic version of the delta method is given. Variance process o"f the estimators and the functionals of the estimators are studied in detail. As an application, a new estimate "for the variance of the product limit estimator is proposed.     

A theorem relating minque and unweighted means estimators of variance components in the one-way design

The unweighted means estimators of variance components in the one-way design are shown to equal the limit of the weighted MINQUE estimators as the ratio of the treatment variance weight to the error variance weight approaches infinity     

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.     

Construction of improved estimators of multinomial proportions

The improved large sample estimation theory for the probabilities of multi¬nomial distribution is developed under uncertain prior information (UPI) that the true proportion is a known quantity. Several estimators based on pretest and the Stein-type shrinkage rules are constructed. The expressions for the bias and risk of the proposed estimators are derived and compared with the maximum likelihood (ml) estimators. It is demonstrated that the shrinkage estimators are superior to the ml estimators. It is also shown that none of the preliminary test and shrinkage estimators dominate each other, though they perform y/ell relative to the ml estimators. The relative dominance picture of the estimators is presented. A simulation study is carried out to assess the performance of the estimators numerically in small samples.     

On multiple change-point estimation for Poisson process

This work is devoted to the problem of change-point parameter estimation in the case of the presence of multiple changes in the intensity function of the Poisson process. It is supposed that the observations are independent inhomogeneous Poisson processes with the same intensity function and this intensity function has two jumps separated by a known quantity. The asymptotic behavior of the maximum-likelihood and Bayesian estimators are described. It is shown that these estimators are consistent, have different limit distributions, the moments converge and that the Bayesian estimators are asymptotically efficient. The numerical simulations illustrate the obtained results.     

CONCOMITANT SCALE ESTIMATION IN REGRESSION PROBLEMS WITH INCREASING DIMENSION

We obtain Bahadur representations for the semi-interquartile range and the median deviation when these estimators are based on the residuals from a linear regression model with increasing dimension. These representations yield a variety of central limit theorems and conditions under which the two estimators are equivalent. In particular, the representations justify the use of the estimators as concomitant scale estimators in general scale equivariant M-estimation of a regression parameter when the dimension of the parameter increases with the sample size.     

New calibration estimator in stratified sampling

This paper develops the theory of calibration estimation and proposes calibration approach alternative to existing calibration estimators for estimating population mean of the study variable using auxiliary variable in stratified sampling. The theory of new calibration estimation is given and optimum weights are derived. A simulation study is carried out to performance of the proposed calibration estimator with other existing calibration estimators. The results reveal that the proposed calibration estimators are more efficient than Tracy et al., Singh et al., Singh calibration estimators of the population mean.     

Variance reduction of estimators arising from Metropolis–Hastings algorithms

The Metropolis–Hastings algorithm is one of the most basic and well-studied Markov chain Monte Carlo methods. It generates a Markov chain which has as limit distribution the target distribution by simulating observations from a different proposal distribution. A proposed value is accepted with some particular probability otherwise the previous value is repeated. As a consequence, the accepted values are repeated a positive number of times and thus any resulting ergodic mean is, in fact, a weighted average. It turns out that this weighted average is an importance sampling-type estimator with random weights. By the standard theory of importance sampling, replacement of these random weights by their (conditional) expectations leads to more efficient estimators. In this paper we study the estimator arising by replacing the random weights with certain estimators of their conditional expectations. We illustrate by simulations that it is often more efficient than the original estimator while in the case of the independence Metropolis–Hastings and for distributions with finite support we formally prove that it is even better than the “optimal” importance sampling estimator.