Efficient and Ad Hoc Estimation in the Bivariate Censoring Model

A large number of proposals for estimating the bivariate survival function under random censoring has been made. In this paper we discuss nonparametric maximum likelihood estimation and the bivariate Kaplan-Meier estimator of Dabrowska. We show how these estimators are computed, present their intuitive background and compare their practical performance under different levels of dependence and censoring, based on extensive simulation results, which leads to a practical advise.

A Note on the Asymptotic Variance of Survival Function in the Semi-Markov Model

In Malani and Neilsen (1992) we have proposed alternative estimates of survival function (for time to disease) using a simple marker that describes time to some intermediate stage in a disease process. In this paper we derive the asymptotic variance of one such proposed estimator using two different methods and compare terms of order 1/n when there is no censoring. In the absence of censoring the asymptotic variance obtained using the Greenwood type approach converges to exact variance up to terms involving 1/n. But the asymptotic variance obtained using the theory of the counting process and results from Voelkel and Crowley (1984) on semi-Markov processes has a different term of order 1/n. It is not clear to us at this point why the variance formulae using the latter approach give different results.

Posterior Simulation in the Generalized Linear Model with Semiparmetric Random Effects

Generalized linear mixed models with semiparametric random effects are useful in a wide variety of Bayesian applications. When the random effects arise from a mixture of Dirichlet process (MDP) model, normal base measures and Gibbs sampling procedures based on the Pólya urn scheme are often used to simulate posterior draws. These algorithms are applicable in the conjugate case when (for a normal base measure) the likelihood is normal. In the non-conjugate case, the algorithms proposed by MacEachern and Müller (1998) and Neal (2000) are often applied to generate posterior samples. Some common problems associated with simulation algorithms for non-conjugate MDP models include convergence and mixing difficulties. This paper proposes an algorithm based on the Pólya urn scheme that extends the Gibbs sampling algorithms to non-conjugate models with normal base measures and exponential family likelihoods. The algorithm proceeds by making Laplace approximations to the likelihood function, thereby reducing the procedure to that of conjugate normal MDP models. To ensure the validity of the stationary distribution in the non-conjugate case, the proposals are accepted or rejected by a Metropolis-Hastings step. In the special case where the data are normally distributed, the algorithm is identical to the Gibbs sampler.