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Continuity of scale parameter estimators with respect to stochastic orders

Bartoszewicz J., Frąszczak M.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 1

57--67

Lehmann and Rojo [8] proposed a concept of invariance of stochastic orders and related probability metrics with respect to increasing transformations of random variables. Bartoszewicz and Benduch [3] and Bartoszewicz and Frąszczak [4] applied a concept of Lehmann and Rojo to new settings. In the paper these results are applied to the problem of robustness in the sense of Zieliński [11], [12]. Metrics related to some stochastic orders are used to study the continuity (robustness) of scale parameter estimators when contaminations of the models are generated by stochastic orders. The exponential model is considered in detail.

Orderings and risk probability functionals in portfolio theory

Ortobelli S., Rachev S., Shalit H., Fabozzi F. J.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 2

203--234

This paper studies and describes stochastic orderings of risk/reward positions in order to define in a natural way risk/reward measures consistent/isotonic to investors’ preferences. We begin by discussing the connection between the theory of probability metrics, risk measures, distributional moments, and stochastic orderings. Then we examine several classes of orderings which are generated by risk probability functionals. Finally, we demonstrate how further orderings could better specify the investor’s attitude toward risk.