Estimating a Quantile of a Symmetric Distribution

Abstract

The problem is to estimate a quantile of a symmetric distribution. The cases of known and unknown center are studied for small and large samples. The estimators for known center are the sample quantile, the symmetrized sample quantile, the sample quantile from flipped over data, the Rao-Blackwellized sample quantile, and a Bayes estimator using a Dirichlet prior. For center unknown, we study the analogues of the first four estimators listed above. For small samples and center known, the Rao-Blackwellized sample quantile performs very well for normal and double exponential distributions while for the Cauchy distribution the flipped over estimator did well. In the center known case, the latter four estimators are asymptotically equivalent, asymptotically optimal in the sense of Hajek's convolution, and asymptotically minimax in the Hajek-LeCam sense. For center unknown, those properties remain true if one uses an adaptive estimator of the center for the symmetrized sample quantile, the flipped over estimator, and the Rao-Blackwell estimator.