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Consider a stochastic process Xn, n = 0, 1, 2, ...such that EXn → x∞ as n → ∞. The sequence {Xn} may be a deterministic one, obtained by using a numerical integration scheme, or obtained from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation but we will assume that we can sample from the distribution of X1, X2, ...Xm for finite m. We propose a scheme for unbiased estimation of the limiting value x∞, together with estimates of standard error and apply this to examples including numerical integrals, root-finding and option pricing in a Heston Stochastic Volatility model.

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In some applications involving regression the values of certain variables are missing by design for some individuals. For example, in two-stage studies (Zhao and Lipsitz, 1992), data on "cheaper" variables are collected on a random sample of individuals in stage I, and then "expensive" variables are measured for a subsample of these in stage II. So the "expensive" variables are missing by design at stage I. Both estimating function and likelihood methods have been proposed for cases where either covariates or responses are missing. We extend the semiparametric maximum likelihood (SPML) method for missing covariate problems (e.g. Chen, 2004; Ibrahim et al., 2005; Zhang and Rockette, 2005, 2007) to deal with more general cases where covariates and/or responses are missing by design, and show that profile likelihood ratio tests and interval estimation are easily implemented. Simulation studies are provided to examine the performance of the likelihood methods and to compare their efficiencies with estimating function methods for problems involving (a) a missing covariate and (b) a missing response variable. We illustrate the ease of implementation of SPML and demonstrate its high efficiency.