Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order
Assume that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X_1,\ldots , X_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> </math> </EquationSource> </InlineEquation> are i.i.d. random variables with a common distribution function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$F$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>F</mi> </math> </EquationSource> </InlineEquation> which precedes a fixed distribution function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$W$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>W</mi> </math> </EquationSource> </InlineEquation> in the convex transform order. In particular, if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$W$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>W</mi> </math> </EquationSource> </InlineEquation> is either uniform or exponential distribution function, then <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$F$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>F</mi> </math> </EquationSource> </InlineEquation> has increasing density and failure rate, respectively. We present sharp upper bounds on the expectations of single order statistics and spacings based on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$X_1,\ldots , X_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> </math> </EquationSource> </InlineEquation>, expressed in terms of the population mean and standard deviation, for the family of all parent distributions preceding <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$W$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>W</mi> </math> </EquationSource> </InlineEquation> in the convex transform order. We also characterize the distributions which attain the bounds, and specify the general results for the distributions with increasing density function. Copyright Springer-Verlag Berlin Heidelberg 2015
