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Peakedness and peakedness ordering

Author

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  • El Barmi, Hammou
  • Mukerjee, Hari

Abstract

The peakedness of a random variable (RV) X about a point a is defined by Pa(x)=P(|X−a|≤x),x≥0. A RV X is said to be less peaked about a than a RV Y about b, denoted by X≤pkd(a,b)Y, if P(|X−a|≤x)≤P(|Y−b|≤x) for all x≥0, i.e., |X−a| is stochastically larger than |Y−b|. These generalize the original definitions of Birnbaum (1948) [2] who considered the cases where X and Y were symmetric about a and b, respectively. Statistical inferences about the distribution functions of continuous X and Y 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.

Suggested Citation

  • El Barmi, Hammou & Mukerjee, Hari, 2012. "Peakedness and peakedness ordering," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 222-233.
    • Handle: RePEc:eee:jmvana:v:111:y:2012:i:c:p:222-233
    DOI: 10.1016/j.jmva.2012.04.006
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    References listed on IDEAS

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    1. Elbarmi, Hammou & Mukerjee, Hari, 2009. "Peakedness and peakedness ordering in symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 594-603, April.
    2. Hammou El Barmi & Hari Mukerjee, 2005. "Inferences Under a Stochastic Ordering Constraint: The k-Sample Case," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 252-261, March.
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