On Kullback–Leibler information of order statistics in terms of the relative risk
The representation of the entropy in terms of the hazard function and its extensions have been studied by many authors including Teitler et al. (IEEE Trans Reliab 35:391–395, <CitationRef CitationID="CR16">1986</CitationRef>). In this paper, we consider a representation of the Kullback–Leibler information of the first <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$r$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>r</mi> </math> </EquationSource> </InlineEquation> order statistics in terms of the relative risk (Park and Shin in Statistics, <CitationRef CitationID="CR14">2012</CitationRef>), the ratio of hazard functions, and extend it to the progressively Type II censored data. Then we study the change in Kullback–Leibler information of the first <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$r$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>r</mi> </math> </EquationSource> </InlineEquation> order statistics according to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$r$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>r</mi> </math> </EquationSource> </InlineEquation> and discuss its relation with Fisher information in order statistics. Copyright Springer-Verlag Berlin Heidelberg 2014
