Quasi-likelihood for median regression models

This article proposes quasi-likelihood equations for median regression models. The quasi-likelihood can be used for dependent observations such as repeated measurements or time series data. To construct a quasi-likelihood equation, we need to specify the relation between the median and the dispersion and also specify the dependency of the observations. If a monotone transformation of the original observation has a Laplace distribution, then the quasi-likelihood is the exact likelihood. Under moderate assumptions, the quasi-likelihood estimates are consistent and have asymptotically normal distributions. The estimates are also shown to have minimal asymptotic variance within a certain class of consistent estimates. The proposed method is illustrated using data from a clinical trial.