Summary: We consider adaptive independent chain (AIC) Metropolis-Hastings algorithms as introduced in a special context by J. Gåsemyr B. Natvig, and E. Sørensen [ibid. 3, No. 1, 51–73 (2001; Zbl 0979.60064)] and developed theoretically by J. Gåsemyr [[On an adaptive version of the Metropolis-Hastings algorithm with independent proposal distribution, Scand. J. Statist. vol. 30, 159–173 (2003)]. The algorithms aim at producing samples from a specific target distribution \(\Pi\) and are adaptive, non-Markovian versions of the Metropolis-Hastings independent chain. A certain parametric class of possible proposal distributions is fixed, and the parameters of the proposal distribution are updated periodically on the basis of the recent history of the chain, thereby obtaining proposals that get ever closer to \(\Pi\). In the former paper a version of these algorithms was shown to be very efficient compared to standard simulation techniques when applied to Bayesian inference in reliability models with at most three dependent parameters.
The aim of the present paper is to investigate the performance of the AIC algorithm when the number of dependent parameters and the complexity of the model increases. As a test case we consider a model treated by E. Arjas and D. Gasbarra [J. Am. Stat. Assoc. 91, No. 435, 1101–1109 (1996; Zbl 0880.62024)]. The target distribution \(\Pi\) is the posterior distribution for the vector \(\mathbf X=(X_1,\dots,X_n)\) of dependent parameters, representing a piecewise constant approximationto the hazard rate \(X(t)\), where \(t_0\leq t\leq t_n\). Especially, for the case \(n=12\) it turned out that some versions of the AIC were very efficientcompared to standard simulation techniques and also to the algorithm applied in Arjas and Gasbarra (1996). This includes a version of the componentwise adaptive independent chain the basic idea of which was given in Gåsemyr (2003).