We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ 1, … , ξ n and a vector of scalars x = (x 1, … , x n ), and 1 ≤ k ≤ n, we provide estimates for $${\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|}$$ and $${\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|}$$ in terms of the values k and the Orlicz norm $${\|y_x\|_M}$$ of the vector y x = (1/x 1, … , 1/x n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ 1|, $${G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}$$ . For example, if ξ 1 is the standard N(0, 1) Gaussian random variable, then $${G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }$$ and $${M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}$$ . We would like to emphasize that our estimates do not depend on the length n of the sequence.
Positivity – Springer Journals
Published: Dec 24, 2010
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