Lehmann, E. L.; Rojo, J. Invariant directional orderings. (English) Zbl 0776.62011 Ann. Stat. 20, No. 4, 2100-2110 (1992). The concept of statistical order of distributions is widely used in different fields of probability theory and statistics. In reliability theory, the distributions are compared in terms of \(G^{-1} F\). This leads to the concept of increasing failure rate, increasing failure rate in average and new better than used. The “stochastic larger” relation is of importance in the theory of testing statistical hypotheses. The monotonicity of the power of monotone rank tests was earlier investigated by the first author [Testing statistical hypotheses. (1959; Zbl 0089.141)].In the present paper preorderings and orderings are studied which are invariant under the group \(\psi\) of monotone transformations. The orderings of monotone likelihood ratio, hazard ordering and stochastic ordering are characterized in terms of the maximal invariant \(k(u)=GF^{-1}(u)\). Orderings in the set of orbits of the group \(\psi\) are used to generate concepts which allow to say when a r.v. \(Y_ 2\) is being further to the right of \(X_ 2\) than \(Y_ 1\) of \(X_ 1\). Reviewer: F.Liese (Rostock) Cited in 2 ReviewsCited in 14 Documents MSC: 62A01 Foundations and philosophical topics in statistics 62E10 Characterization and structure theory of statistical distributions 62N05 Reliability and life testing 60E05 Probability distributions: general theory Keywords:directional orderings of univariate distributions; characterizations; location families; distance between distribution functions; stochastic larger relation; preorders; reliability; increasing failure rate; increasing failure rate in average; new better than used; monotonicity; power of monotone rank tests; monotone transformations; orderings of monotone likelihood ratio; hazard ordering; stochastic ordering; maximal invariant Citations:Zbl 0089.141 PDFBibTeX XMLCite \textit{E. L. Lehmann} and \textit{J. Rojo}, Ann. Stat. 20, No. 4, 2100--2110 (1992; Zbl 0776.62011) Full Text: DOI