Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternative approach is to approximate the intractable integrals using Monte Carlo averages. Several dierent algorithms based on this idea have been proposed. In this paper we discuss the relative merits of simulated maximum likelihood, Monte Carlo EM, Monte Carlo Newton-Raphson and stochastic approximation. Key words and phrases : Eciency, Monte Carlo EM, Monte Carlo Newton-Raphson, Rate of convergence, Simulated maximum likelihood, Stochastic approximation All three authors partially supported by NSF Grant DMS-00-72827. 1 1

To account for the input-model and input-parameter uncertainties inherent in many simulations as well as the usual stochastic uncertainty, we present a Bayesian input-modeling technique that yields improved point and confidence-interval estimators for a selected posterior mean response. Exploiting prior information to specify the prior probabilities of the postulated input models and the associated prior input-parameter distributions, we use sample data to compute the posterior input-model and input-parameter distributions. Our Bayesian simulation replication algorithm involves: (i) estimating parameter uncertainty by randomly sampling the posterior input-parameter distributions; (ii) estimating stochastic uncertainty by running independent replications of the simulation using each set of input-model parameters sampled in (i); and (iii) estimating input-model uncertainty by weighting the responses generated in (ii) using the corresponding posterior input-model probabilities. Sampling effort is allocated among input models to minimize final point-estimator variance subject to a computing-budget constraint. A queueing simulation demonstrates the advantages of this approach.

Abstract not found

this paper, however, we will concentrate mainly on business surveys, which are typically characterised by more straightforward designs (often single stage stratified designs), but may use estimators which have more variable weights, such as ratio and regression estimators (Cochran 1977). These estimators are typically more accurate because there are larger differences in the sizes of businesses which are captured and compensated for by using an appropriate model.

with 1 or 2 uppermost, S 2 if it lands with 3 or 4 uppermost, or S 3 if it lands with 5 or 6 uppermost. Note that in general the set of possible samples need not consist of all possible samples of a given size (and indeed it can be useful to consider cases where the sample size is random), and not all possible samples need have the same probability. Once one has a method for finding a sample, we need to have a method of computing an estimate of any quantity of interest. For example, one could take the mean of the specimens in the sample. Alternatives to Random Sampling There are various alternative ways in which a sample can be taken. For example, take the most easily obtainable specimens. The pitfalls in such a procedure, which can be referred to as accessibility or haphazard sampling, are obvious in that such a sample is unlikely to be any any real sense `representative'

ernative ways in which a sample can be taken. For example, take the most easily obtainable specimens. The pitfalls in such a procedure, which can be referred to as accessibility or haphazard sampling, are obvious in that such a sample is unlikely to be any any real sense `representative'. A second method would be to number the specimens in some more or less systematic manner and then take every nth specimen for some suitable value of n, which can be referred to as a systematic sample. However, there are warnings about its use to be garnered from section 5.2 of Gray and Gee (1972). The 1966 sample census attempted to use a systematic sample for caravan sites and for hospitals, schools, etc. The sort of thing that went wrong was that on one caravan site where the random start was 8, the enumerator took the 8th, 16th, 24th, : : : caravans, instead of the 8th, 18th, 28th, : : : (i.e. took the random start as the sampling interval). In the hospital and school records, interviewers were to

Abstract not found

doi:10.1111/j.1469-7998.2006.00055.x

Aerial surveys of belugas, or white whales, Delphinapterus leucas, were conducted off

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