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Abstract: The peakedness of a random variable (RV) about a point is defined by . A RV is said to be less peaked about than a RV about , denoted by , if for all , i.e., is stochastically larger than . These generalize the original definitions of Birnbaum (1948) who considered the cases where and were symmetric about and , respectively. Statistical inferences about the distribution functions of continuous and under peakedness ordering in the symmetric case have been treated in the literature. Rojo et al. (2007) 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. [Copyright &y& Elsevier]

Abstract: There are many ways to measure the dispersion of a random variable. One such method uses the concept of peakedness. If the random variable is symmetric about a point , then Birnbaum [Z.W. Birnbaum, On random variables with comparable peakedness, The Annals of Mathematical Statistics 19 (1948) 76–81] defined the function , as the peakedness of . If two random variables, and , are symmetric about the points and , respectively, then is said to be less peaked than , denoted by , if for all , i.e.,  is stochastically larger than . For normal distributions this is equivalent to variance ordering. Peakedness ordering can be generalized to the case where and 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. [Copyright &y& Elsevier]

Abstract: In competing risks studies, the Kaplan–Meier estimators of the distribution functions (DFs) of lifetimes and the corresponding estimators of cumulative incidence functions (CIFs) are used widely when no prior information is available for these distributions. In some cases better estimators of the DFs of lifetimes are available when they obey some inequality constraints, e.g., if two lifetimes are stochastically or uniformly stochastically ordered, or some functional of a DF obeys an inequality in an empirical likelihood estimation procedure. If the restricted estimator of a lifetime differs from the unrestricted one, then the usual estimators of the CIFs will not add up to the lifetime estimator. In this paper we show how to estimate the CIFs in this case. These estimators are shown to be strongly uniformly consistent. In all cases we consider, when the inequality constraints are strict the asymptotic properties of the restricted and the unrestricted estimators are the same, thus providing the asymptotic properties of the restricted estimators essentially “free of charge”. We give an example to illustrate our procedure. [Copyright &y& Elsevier]