Utilizing observations and ranks in kernel estimation

Abstract

For univariate i.i.d. samples, analyses are usually performed on observations themselves, or on the ranks of the observations when they are ordered. This is true in both the parametric and non-parametric setting. However, observations and ranks can be utilized simultaneously. This work concentrates on developing methods that combine observations and ranks in kernel-based estimation methods. The standard kernel density estimator, transformation kernel density estimator, and the Nadaraya-Watson kernel regression estimator are modified using ranks. Excluding the modified transformation kernel density case, the asymptotic properties are investigated for the estimators. For all estimators, various bandwidth selection methods are explored. Simulation is used to compare the modified estimators to the standard estimators. It is demonstrated that when samples are taken from skewed distributions, the modified kernel density estimator outperforms the standard estimator. Additionally, the modified Nadaraya-Watson outperforms the standard estimator in a variety of cases. The modified transformation density estimator did not outperform the unmodified estimator.