Peakedness and peakedness ordering

The peakedness of a random variable (RV) XX about a point aa is defined by Pa(x)=P(|X−a|≤x),x≥0. A RV XX is said to be less peaked about aa than a RV YY about bb, denoted by X≤pkd(a,b)YX≤pkd(a,b)Y, if P(|X−a|≤x)≤P(|Y−b|≤x)P(|X−a|≤x)≤P(|Y−b|≤x) for all x≥0x≥0, i.e., |X−a||X−a| is stochastically larger than |Y−b||Y−b|. These generalize the original definitions of Birnbaum (1948) [2] who considered the cases where XX and YY were symmetric about aa and bb, respectively. Statistical inferences about the distribution functions of continuous XX and YY under peakedness ordering in the symmetric case have been treated in the literature. Rojo et al. (2007) [12] provided estimators of the distributions in the general case and analyzed their properties. We show that these estimators could have poor asymptotic properties relative to those of the empiricals. We provide improved estimators of the DFs, show that they are consistent, derive the weak convergence of the estimators, compare them with the empirical estimators, and provide formulas for statistical inferences. An example is also used to illustrate our theoretical results.