Method of Quantiles

The underlying idea of Reich, Fuentes, and Dunson (2009) to use the asymptotic normal approximation of the quantile regression estimator as a “substitute” likelihood can be regarded as a convenient dumbing-down of the Jeffrey’s idea elaborated by Lavine (1995) and Dunson and Taylor (2005). The obvious disadvantage of the original Jeffrey’s suggestion is that it is difficult to compute/update the binomial proposal, whereas the normal approximation is made for Bayes rule. One way to enter the dungeons and dragons of Reich, Fuentes, and Dunson (2009) is to peel away all the Bayesian prior layers and focus on the basic model in its simplest setting. This puts us in the realm of MoQ, or method of quantiles. Essentially, MoQ is the much beloved MoM, with moments replaced by quantiles. Quantiles are moments too, of a sort. Suppose we have a parametric model: Yi ∼ f(y, θ), having quantiles q(τ, θ) for τ ∈ (0, 1). A vector of sample quantiles q̂n = (q̂n(τi) based on a random sample of size n, is asymptotically Gaussian with mean q(θ) = (q(τi, θ)) and covariance matrix V = τi ∧ τj − τiτj f(q(τi, θ))f(q(τj , θ)) Thus, a natural estimator of θ is