Asymptotically optimal estimation of smooth functionals for interval censoring, part 1

Estimation in the interval censoring model is considered. A class of smooth functionals is introduced, of which the mean is an example. The asymptotic information lower bound for such functionals can be represented as an inner product of two functions. In case 1, i.e. one observation time per unobservable event time, both functions can be given explicitly. We mainly consider case 2, with two observation times for each unobservable event time, in the situation that the observation times can not become arbitrarily close to each other. For case 2, one of the functions in the inner product can only be given implicitly as solution to a Fredholm integral equation. We study properties of this solution and, in a sequel to this paper, prove that the nonparametric maximum likelihood estimator of the functional asymptotically reaches the information lower bound.     

ML– and semiparametric estimation in logistic models with incomplete covariate data

ML–estimation of regression parameters with incomplete covariate information usually requires a distributional assumption regarding the concerned covariates that implies a source of misspecification. Semiparametric procedures avoid such assumptions at the expense of efficiency. In this paper a simulation study with small sample size is carried out to get an idea of the performance of the ML–estimator under misspecification and to compare it with the semiparametric procedures when the former is based on a correct assumption. The results show that there is only a little gain by correct parametric assumptions, which does not justify the possibly large bias when the assumptions are not met. Additionally, a simple modification of the complete case estimator appears to be nearly semiparametric efficient.     

Asymptotically optimal estimation of smooth functionals for interval censoring, part 2

Estimation in the interval censoring model is considered. A class of smooth functionals is introduced, of which the mean is an example. We consider case 2, with two observation times for each unobservable event time, in the situation that the observation times cannot become arbitrarily close to each other. It is proved that the nonparametric maximum likelihood estimator of the functional asymptotically reaches the information lower bound.     

Density estimation in the uniform deconvolution model

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data.     

A true simulation study of three estimators at equal protection of respondents in randomized response sampling

In this study, three different estimators for estimating the proportion of a sensitive attribute in survey sampling are compared at equal protection of the respondents. The three estimators considered are due to Odumade and Singh (2009, Comm. Statist. Theory Methods) , Singh and Sedory (2011, Sociological Methods and Research) and a new estimator obtained by minimizing a chi‐squared distance. A SAS Macro is developed to compare these three estimators using a simulation study at equal protection of the respondents. A set of data from a real face‐to‐face interview was collected using two decks of cards and has been analyzed. The results are discussed.     

Asymptotic linearity of minimax estimators

The celebrated local asymptotic minimax (LAM) theorem due to HÁjek (1972) also includes the statement that a LAM estimator Is necessarily asymptotically linear. A similar result. is true for semi-parametric models, but Hájek's result doesn't apply to this case as the efficient influence function is often not contained in the (proper) tangent space. This note gives a simple, elementary proof of both the LAM theorem and the necessity of asymptotic linearity of a LAM estimator sequence.     

Some model-restricted shrinkage estimators for contingency tables with missing data

For contingency tables with extensive missing data, the unrestricted MLE under the saturated model, computed by the EM algorithm, is generally unsatisfactory. In this case, it may be better to fit a simpler model by imposing some restrictions on the parameter space. Perlman and Wu (1999) propose lattice conditional independence (LCI) models for contingency tables with arbitrary missing data patterns. When this LCI model fits well, the restricted MLE under the LCI model is more accurate than the unrestricted MLE under the saturated model, but not in general. Here we propose certain empirical Bayes (EB) estimators that adaptively combine the best features of the restricted and unrestricted MLEs. These EB estimators appear to be especially useful when the observed data is sparse, even in cases where the suitability of the LCI model is uncertain. We also study a restricted EM algorithm (called the ER algorithm) with similar desirable features. Received: July 1999     

Nonparametric estimators of the bivariate survival function under random censoring

A large number of proposals for estimating the bivariate survival function under random censoring have been made. In this paper we discuss the most prominent estimators, where prominent is meant in the sense that they are best for practical use; Dabrowska's estimator, the Prentice–Cai estimator, Pruitt's modified EM-estimator, and the reduced data NPMLE of van der Laan. We show how these estimators are computed and present their intuitive background. The asymptotic results are summarized. Furthermore, we give a summary of the practical performance of the estimators under different levels of dependence and censoring based on extensive simulation results. This leads also to a practical advise.     

Smooth estimators for estimating order restricted scale parameters of two gamma distributions

We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation.     

Estimation of multiple poisson means: A compromise between maximum likelihood and empirical bayes estimators

For estimatingp(⩾ 2) independent Poisson means, the paper considers a compromise between maximum likelihood and empirical Bayes estimators. Such compromise estimators enjoy both good componentwise as well as ensemble properties. Research supported by the NSF Grant Number MCS-8218091.     

On finite sample properties of alternative estimators of coefficients in a structural equation with many instruments

We compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases. We have found that the LIML estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large, that is, the micro-econometric models with “many instruments” in the terminology of recent econometric literature.     

Shrinkage estimation of the exponentiated Weibull regression model for time-to-event data

The exponentiated Weibull distribution is a convenient alternative to the generalized gamma distribution to model time-to-event data. It accommodates both monotone and nonmonotone hazard shapes, and flexible enough to describe data with wide ranging characteristics. It can also be used for regression analysis of time-to-event data. The maximum likelihood method is thus far the most widely used technique for inference, though there is a considerable body of research of improving the maximum likelihood estimators in terms of asymptotic efficiency. For example, there has recently been considerable attention on applying James–Stein shrinkage ideas to parameter estimation in regression models. We propose nonpenalty shrinkage estimation for the exponentiated Weibull regression model for time-to-event data. Comparative studies suggest that the shrinkage estimators outperform the maximum likelihood estimators in terms of statistical efficiency. Overall, the shrinkage method leads to more accurate statistical inference, a fundamental and desirable component of statistical theory.     

Estimation of P[Y < X] for generalized exponential distribution

This paper deals with the estimation of P[Y < X] when X and Y are two independent generalized exponential distributions with different shape parameters but having the same scale parameters. The maximum likelihood estimator and its asymptotic distribution is obtained. The asymptotic distribution is used to construct an asymptotic confidence interval of P[Y < X]. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator and Bayes estimator of P[Y < X] are obtained. Different confidence intervals are proposed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a simulated data set has also been presented for illustrative purposes.Part of the work was supported by a grant from the Natural Sciences and Engineering Research Council     

Efficiency Considerations in the Additive Hazards Model with Current Status Data

For current status data, L IN , O AKES and Y ing (1998) proposed a procedure for estimation of the regression parameters in the additive hazards model that makes clever use of martingale theory. However, one of the outstanding problems posed in the paper was the issue of efficient estimation, as their estimators do not attain the semiparametric information bound. In this paper, we explore this issue and provide a characterization of the NPMLE. We conduct efficiency comparisons between the NPMLE and the procedure of L IN et al . (1998) analytically and numerically through analysis of a dataset from a tumorigenicity experiment.     

Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models

The limit distribution of the quasi-maximum likelihood estimator (QMLE) for parameters in the ARMA-GARCH model remains an open problem when the process has infinite 4th moment. We propose a self-weighted QMLE and show that it is consistent and asymptotically normal under only a fractional moment condition. Based on this estimator, the asymptotic normality of the local QMLE is established for the ARMA model with GARCH (finite variance) and IGARCH errors. Using the self-weighted and the local QMLEs, we construct Wald statistics for testing linear restrictions on the parameters, and their limiting distributions are given. In addition, we show that the tail index of the IGARCH process is always 2, which is independently of interest.     

Asymptotic normality of the NPMLE of linear functionals for interval censored data, case 1

We give a new proof of the asymptotic normality of a class of linear functionals of the nonparametric maximum likelihood estimator (NPMLE) of a distribution function with "case 1" interval censored data. In particular our proof simplifies the proof of asymptotic normality of the mean given in Groeneboom and Wellner (1992). The proof relies strongly on a rate of convergence result due to van de Geer (1993), and methods from empirical process theory.     

Comparison of tail index estimators

We compare various estimators for the index of distribution functions with regularly varying tails by calculating their asymptotic mean squared errors after choosing the optimal number of upper order statistics involved (which is different for different estimators).     

On Pooling of Data and Its Relative Efficiency

Pooling of data is often carried out to protect privacy or to save cost, with the claimed advantage that it does not lead to much loss of efficiency. We argue that this does not give the complete picture as the estimation of different parameters is affected to different degrees by pooling. We establish a ladder of efficiency loss for estimating the mean, variance, skewness and kurtosis, and more generally multivariate joint cumulants, in powers of the pool size. The asymptotic efficiency of the pooled data non‐parametric/parametric maximum likelihood estimator relative to the corresponding unpooled data estimator is reduced by a factor equal to the pool size whenever the order of the cumulant to be estimated is increased by one. The implications of this result are demonstrated in case–control genetic association studies with interactions between genes. Our findings provide a guideline for the discriminate use of data pooling in practice and the assessment of its relative efficiency. As exact maximum likelihood estimates are difficult to obtain if the pool size is large, we address briefly how to obtain computationally efficient estimates from pooled data and suggest Gaussian estimation and non‐parametric maximum likelihood as two feasible methods.     

Identifying, estimating and testing restricted cointegrated systems: An overview

The notion of cointegration has led to a renewed interest in the identification and estimation of structural relations among economic time series. This paper reviews the different approaches that have been put forward in the literature for identifying cointegrating relationships and imposing (possibly over-identifying) restrictions on them. Next, various algorithms to obtain (approximate) maximum likelihood estimates and likelihood ratio statistics are reviewed, with an emphasis on so-called switching algorithms. The implementation of these algorithms is discussed and illustrated using an empirical example.     

On the analysis of multivariate growth curves

Growth curve data arise when repeated measurements are observed on a number of individuals with an ordered dimension for occasions. Such data appear frequently in almost all fields in which statistical models are used, for instance in medicine, agriculture and engineering. In medicine, for example, more than one variable is often measured on each occasion. However, analyses are usually based on exploration of repeated measurements of only one variable. The consequence is that the information contained in the between-variables correlation structure will be discarded.  In this study we propose a multivariate model based on the random coefficient regression model for the analysis of growth curve data. Closed-form expressions for the model parameters are derived under the maximum likelihood (ML) and the restricted maximum likelihood (REML) framework. It is shown that in certain situations estimated variances of growth curve parameters are greater for REML. Also a method is proposed for testing general linear hypotheses. One numerical example is provided to illustrate the methods discussed. Received: 22 February 1999