Efficiency of a Linear Combination of the Median and the Sample Mean: The Double Truncated Normal Distribution

Abstract

We characterize all symmetric location models for which a linear combination of the median and the sample mean is an asymptotically efficient estimator of the location parameter. The resulting model can be understood as a symmetrized or double truncated normal distribution. A simple algorithm to estimate the parameters is given and an application is presented. Copyright 2004 Board of the Foundation of the Scandinavian Journal of Statistics..

The mixture of left–right truncated normal distributions

“…(3.2) is the symmetrized truncated normal considered in Damilano and Puig (2004). In the general case, 0 < < 1, the support of the distribution is R. If = 0 or 1, then the support is a half straight line.…”

“…It is proved that there is only one maximum for the likelihood function and the maximum is closely related to the coefficient of variation. The results apply, in particular, to the symmetric mixture of truncated distributions characterized by Damilano and Puig (2004), in terms of the asymptotic efficiency of the median and the sample mean. Continuous mixed truncated normal has been used to generate asymmetric models (see Pedersen and Hwang, 2007).…”

Robust Mean Estimation with the Bayesian Median of Means

“…The idea of combining mean and median estimators has been visited several times in the statistical robustness literature, particularly for the estimation of location parameters in symmetric distribu- tions. For instance, [Lai et al, 1983] propose an adaptive estimator that picks either the sample mean or median to estimate the center of a symmetric distribution, while [Damilano and Puig, 2004] and [Chan and He, 1994] investigate using linear combinations of mean and medians, with weights picked according to asymptotic criteria. More recently, there has been intense work on the so-called median of means estimator (see [Alon et al, 1999], [Jerrum et al, 1986], [Nemirovsky and Yudin, 1983]).…”

An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator

“…This idea of taking an affine combination of two unbiased estimators to define a new unbiased estimator appeared at least 200 years ago in a work by Laplace (see Stigler 1973 ), and is still found in modern use (e.g., see recent work by Damilano and Puig 2004 ).…”