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Markov chain Monte Carlo (MCMC) methods provide an important means to simulate from almost any probability density. To approximate non-standard discrete distributions, the equation-solving MCMC estimator was developed as an alternative to the classical frequency estimator. The used simulation scheme is the Metropolis–Hastings (M–H) algorithm. Recently, this estimator has been extended to the specific context of 2-step Metropolis-Hastings with delayed rejection (MHDR) algorithm, which allowed a considerable reduction in asymptotic variance. In this paper, we propose an adaptation of equation-solving estimator to the case of general n-step MHDR sampler. The aim is to further improve the precision. An application to a Bayesian hypothesis test problem shows the high performance, in terms of accuracy, of the equation-solving estimator, based on a MHDR algorithm with more than two stages. [ABSTRACT FROM AUTHOR]