Peakedness and peakedness ordering in symmetric distributions

There are many ways to measure the dispersion of a random variable. One such method uses the concept of peakedness. If the random variable XX is symmetric about a point μμ, then Birnbaum [Z.W. Birnbaum, On random variables with comparable peakedness, The Annals of Mathematical Statistics 19 (1948) 76–81] defined the function Pμ(x)=P(|X−μ|≤x),x≥0, as the peakedness of XX. If two random variables, XX and YY, are symmetric about the points μμ and νν, respectively, then XX is said to be less peaked than YY, denoted by X≤pkd(μ,ν)YX≤pkd(μ,ν)Y, if P(|X−μ|≤x)≤P(|Y−ν|≤x)P(|X−μ|≤x)≤P(|Y−ν|≤x) for all x≥0x≥0, i.e., |X−μ||X−μ| is stochastically larger than |Y−ν||Y−ν|. For normal distributions this is equivalent to variance ordering. Peakedness ordering can be generalized to the case where μμ and νν are arbitrary points. However, in this paper we study the comparison of dispersions in two continuous random variables, symmetric about their respective medians, using the peakedness concept where normality, and even moment assumptions are not necessary. We provide estimators of the distribution functions under the restriction of symmetry and peakedness ordering, 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 given to illustrate the theoretical results.