Efficient Robbins–Monro procedure for multivariate binary data

This paper considers the problem of jointly estimating marginal quantiles of a multivariate distribution. A sufficient condition for an estimator that converges in probability under a multivariate version of Robbins–Monro procedure is provided. We propose an efficient procedure which incorporates the correlation structure of the multivariate distribution to improve the estimation especially for cases involving extreme marginal quantiles. Estimation efficiency of the proposed method is demonstrated by simulation in comparison with a general multivariate Robbins–Monro procedure and an efficient Robbins–Monro procedure that estimates the marginal quantiles separately.