On estimation of peakedness-ordered distributions

A random variable X is said to be less peaked about a point μ than a random variable Y about a point ν, denoted by X⪯pkd(μ, ν)Y, if P(|X-μ|≤x)≤P(|Y-ν|≤x),∀x≥0 \begin{eqnarray*} P(|X-\mu | \le x) \le P(|Y-\nu | \le x), \quad \forall x \ge 0 \end{eqnarray*} This paper develops new consistent estimators of the distribution functions of two continuous random variables X and Y when X is less peaked than Y. Through simulation studies, we show that our estimators outperform in terms of mean-squared error the estimators proposed in El Barmi and Mukerjee (2012) for the same problem. To illustrate the theory, an empirical example is presented.