Estimation of the innovation quantile density function of an AR(p) process based on autoregression quantiles

Abstract

In this paper, we propose two types of estimator (one of histogram type, the other a kernel estimate) of the quantile density (or sparsity) function α\mapsto [f(F^-1(α ))]^-1 associated with the innovation density f of an autoregressive model of order p. Our estimators are based on autoregression quantiles. Contrary to more classical estimators based on estimated residuals, they are autoregression-invariant and scale-equivariant. Their asymptotic behaviour is derived from a uniform Bahadur representation for autoregression quantiles - a result of independent interest. Simulations are carried out to illustrate their performance.