Nonparametric density estimation based on the truncated mean

Motivated by the optimality condition of a quantile loss minimization problem, a new family of closed-form density estimators based on truncated means is developed and found to achieve smaller mean squared errors in estimating the tails of the normal and gamma distributions when compared to the symmetric Rosenblatt–Parzen kernel estimator.

► A density function can be expressed via the first derivative of the truncated mean.
► A new family of closed-form density estimators is derived from this relationship.
► These estimators have the same limiting variance as the Rosenblatt kernel estimators.
► But their asymptotic bias formulas differ.
► The new estimators are better for estimating the tails of some common distributions.