This paper is an expository survey of the main methods that have been developed for numerical evaluation of multiple integrals. Among the approaches discussed are: the Monte Carlo method and its generalizations; number-theoretical methods, based essentially on the ideas of Diophantine approximation and equidistribution modulo 1; the functional analysis approach, in which the quadrature error is regarded as a linear functional and one attempts to minimize its norm; and the classical approach of designing formulas to be exact for polynomials of high degree while using as few values of the integrand as possible. Most of the research in this field is quite recent.

Numerical Evaluation of Multiple Integrals

We consider the first steps towards implementing capture–recapture mixed models (CR2Ms) in program E-SURGE. The main issue when estimating the parameters of mixed models is that integrals associated with the random effects distributions need to be dealt with. Rather than using a Bayesian approach with Markov chain Monte Carlo and in line with Gimenez and Choquet (Ecology 91:951–957, 2010), we show that a frequentist approach using numerical integration can be tractable when independent clusters of individuals can be identified. In this case, the maximum likelihood approach is time-efficient because the dimension of the integral for the likelihood is small. This allows us to integrate the likelihood by an efficient and appropriate quadrature method with a procedure for error control. Building on program E-SURGE (Choquet et al. in Modeling demographic processes in marked populations, volume 3 of Springer series: environmental and ecological statistics. Springer, Dunedin, 2009b), we extend the GEMACO language (Choquet in Can J Stat 36:43–57, 2008) to incorporate random effects in a large set of capture–recapture models, including multievent models (Pradel in Biometrics 61:442–447, 2005). To illustrate the flexible implementation of CR2Ms in E-SURGE, we consider two real examples, one with an individual random effect and one with group random effects. Future developments and limitations are also discussed.

Towards built-in capture–recapture mixed models in program E-SURGE

The Bernstein-type approximation using the beta-binomial distribution is proposed and studied. Both univariate and multivariate Bernstein-type approximations are studied. The uniform convergence and the degree of approximation are studied. The Bernsteintype estimators of smooth functions of population means are also proposed and studied.

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Bernstein-type approximation using the beta-binomial distribution

The method of Ermakov and Zolotukhin is discussed along with its later developments. By introducing the idea of pseudo-implementation a practical assessment of the method is made. The performance of the method is found to be unimpressive in comparison with a recent regression method.

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On the method of Ermakov and Zolotukhin for multiple integration

On "closed" normal series and generalized Bernstein polynomials

Two variance reducing techniques, the method of control variates and the method of importance sampling, which are used in increasing the efficiency of a Monte Carlo evaluation of an integral, are described. It is shown under very general conditions that the Bernstein polynomials can serve as both a control variate and an importance sampling function. In addition these methods can also be combined to further increase the efficiency of the Monte Carlo evaluation. An example is given to illustrate the method.

Lloyd Rosenberg

Papers

Bernstein Polynomials and Monte Carlo Integration