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Incomplete information on explanatory variables is commonly encountered in studies of possibly censored event times. A popular approach to deal with partially observed covariates is multiple imputation, where a number of completed data sets, that can be analyzed by standard complete data methods, are obtained by imputing missing values from an appropriate distribution. We show how the combination of multiple imputations from a compatible model with suitably estimated parameters and the usual Cox regression estimators leads to consistent and asymptotically Gaussian estimators of both the finite-dimensional regression parameter and the infinite-dimensional cumulative baseline hazard parameter. We also derive a consistent estimator of the covariance operator. Simulation studies and an application to a study on survival after treatment for liver cirrhosis show that the estimators perform well with moderate sample sizes and indicate that iterating the multiple-imputation estimator increases the precision.

In many stated choice experiments researchers observe the random variablesVt,Xt, andYt= 1{U+δ⊤Xt+ εt

In this paper we investigate a class of semiparametric models for panel datasets where the cross-section and time dimensions are large. Our model contains a latent time series that is to be estimated and perhaps forecasted along with a nonparametric covariate effect. Our model is motivated by the need to be flexible with regard to functional form of covariate effects but also the need to be practical with regard to forecasting of time series effects. We propose estimation procedures based on local linear kernel smoothing; our estimators are all explicitly given. We establish the pointwise consistency and asymptotic normality of our estimators. We also show that the effects of estimating the latent time series can be ignored in certain cases.

Summary Multiple imputation has become viewed as a general solution to missing data problems in statistics. However, in order to lead to consistent asymptotically normal estimators, correct variance estimators and valid tests, the imputations must be proper. So far it seems that only Bayesian multiple imputation, i.e. using a Bayesian predictive distribution to generate the imputations, or approximately Bayesian multiple imputations has been shown to lead to proper imputations in some settings. In this paper, we shall see that Bayesian multiple imputation does not generally lead to proper multiple imputations. Furthermore, it will be argued that for general statistical use, Bayesian multiple imputation is inefficient even when it is proper. Resume On a pris l'habitude de considerer l'imputation multiple comme une solution generale au probleme des donnees manquantes en statistique. Cependant, afin d'obtenir des estimateurs coherents et asymptotiquement normaux, des estimations correctes de la variance et des tests valides, il faut que les imputations soient approprees. Jusqu'alors, il semble que c'est seulement dans le cas de l'imputation multiple bayesienne, c'esta-dire utilisant une distribution predictive bayesienne pour generer les imputations, ou de ses approximations, que l'on a montre que l'on obtenait des imputations appropriees dans certains cas particuliers. Dans cet article, on montre qu'en general l'imputation multiples bayesienne ne conduit pas a des imputations multiples appropriees. De plus, on argumente l'idee selon laquelle, de maniere generale, l'imputation multiple bayesienne est inefficace meeme si elle appropree.

In this paper, uniform consistency of estimators obtained from kernel-based local linear estimating equations is derived. Furthermore, it is shown that the rate of convergence is at least n 1/3 for a suitable choice of bandwidth. These results are used to find asymptotic results for the integral of the estimator. As an application, we consider an ‘inverse probability of missingness reweighted estimating equation’ estimator.

Imputation is a much used method for handling missing data. It is appealing as it separates the missing data part of the analysis, which is handled by imputation, and the estimation part, which is handled by complete data methods. Most imputation methods, however, either rely on strict parametric assumptions or are rather ad hoc in which case they often only work approximately under even stricter assumptions. In this paper a non-parametric imputation method is proposed. Since it is non-parametric it works under quite general assumptions. In particular, a model for the complete data is not required in the imputation step, and the complete data method used after the imputation may be a general estimating equation for estimating a finite-dimensional parameter. Large sample results for the resulting estimator are given.

The EM algorithm is a popular and useful algorithm for finding the maximum likelihood estimator in incomplete data problems. Each iteration of the algorithm consists of two simple steps: an E-step, in which a conditional expectation is calculated, and an M-step, where the expectation is maximized. In some problems, however, the EM algorithm cannot be applied since the conditional expectation required in the E-step cannot be calculated. Instead the expectation may be estimated by simulation. We call this a simulated EM algorithm. The simulations can, at least in principle, be done in two ways. Either new independent random variables are drawn in each iteration, or the same uniforms are re-used in each iteration. In this paper the properties of these two versions of the simulated EM algorithm are discussed and compared.

Many types of data are often incompletely observed. How incompletely is typically randomly determined. Heitjan and Rubin (Annals of Statistics, 1991) proposed a condition, “coarsened at random” or CAR, ensuring ignorability of this randomness in discrete sample spaces. In general sample spaces CAR comes in two flavors according to whether it is defined in terms of probabilities or densities. In this paper, CAR defined in terms of densities, called relative CAR, is discussed as a condition for ignorability in a statistical model allowing for partial observation of random elements determining the degree of incompleteness in the observation of the data.

In Rubin (1976) the missing at random (MAR) and missing completely at random (MCAR) conditions are discussed. It is concluded that the MAR condition allows one to ignore the missing data mechanism when doing likelihood or Bayesian inference but also that the stronger MCAR condition is in some sense the weakest generally sufficient condition allowing (conditional) frequentist inference while ignoring the missing data mechanism. In this paper it is shown that (a slightly strengthened version of) the MAR condition is sufficient to yield ordinary large sample results for estimators and test statistics and thus may be used for (asymptotic) frequentist inference.

Introduction to general and generalized linear models, by Henrik Madsen and Poul Thyregod, Boca Raton, Chapman & Hall/CRC Press, 2011, xii+302 pp., £ 39.99 or US$83.95 (hardback), ISBN 978-1-420-09...

The EM algorithm is a much used tool for maximum likelihood estimation in missing or incomplete data problems. However, calculating the conditional expectation required in the E-step of the algorithm may be infeasible, especially when this expectation is a large sum or a high-dimensional integral. Instead the expectation can be estimated by simulation. This is the common idea in the stochastic EM algorithm and the Monte Carlo EM algorithm.