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An Lp-view of the Bahadur-Kiefer Theorem

Periodica Mathematica Hungarica (impact factor: 0.26). 07/2005; 50(1):79-98. DOI:10.1007/s10998-005-0004-x pp.79-98

ABSTRACT Let ]] > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /equationsource > < /inlineequation > ]] > ]] > ]] > ]] > ]] > ]] > ]] > ]] > ]] > ]] > ]] > an]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>\alpha_n and bn\beta_n be respectively the uniform empirical and quantile processes, and define Rn = an + bnR_n = \alpha_n + \beta_n, which usually is referred to as the Bahadur--Kiefer process. The well-known Bahadur-Kiefer theorem confirms the following remarkable equivalence: $\|R_n\| /\sqrt{\| \alpha_n \| }\, \sim \, n^{-1/4} (\log n)^{1/2}$\|R_n\| /\sqrt{\| \alpha_n \| }\, \sim \, n^{-1/4} (\log n)^{1/2} almost surely, as nn goes to infinity, where || f|| = sup0 £ t £ 1 |f(t)|\| f\| =\sup_{0\le t\le 1} |f(t)| is the L¥L^\infty-norm. We prove that ||Rn||2 /Ö{|| an ||1} ~ n-1/4\|R_n\|_2 /\sqrt{\| \alpha_n \|_1}\, \sim \, n^{-1/4} almost surely, where || ||p\| \, \cdot \, \|_p is the LpL^p-norm. It is interesting to note that there is no longer any logarithmic term in the normalizing function. More generally, we show that n1/4 ||Rn||p /Ö{|| an ||(p/2)}n^{1/4} \|R_n\|_p /\sqrt{\| \alpha_n \|_{(p/2)}} converges almost surely to a finite positive constant whose value is explicitly known.

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Keywords

define Rn

logarithmic term

LpL^p-norm

nn

quantile processes

uniform empirical