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Abstract <p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\hat{m}_{n}\)</EquationSource> <!--MMStat2670003Ferger-m1--> </InlineEquation> be an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(M\)</EquationSource> <!--MMStat2670003Ferger-m2--> </InlineEquation>-estimator for a parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(m\)</EquationSource> <!--MMStat2670003Ferger-m3--> </InlineEquation> based on a sample of size <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n\in\mathbb{N}\)</EquationSource> <!--MMStat2670003Ferger-m4--> </InlineEquation>. We derive exponential and polynomial upper bounds for the tail-probabilities of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\sup_{k\geq n}|\hat{m}_{k}-m|\)</EquationSource> <!--MMStat2670003Ferger-m5--> </InlineEquation> according as a boundedness- or a moment-condition is fulfilled. This enables us to derive rates of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(r\)</EquationSource> <!--MMStat2670003Ferger-m6--> </InlineEquation>-complete convergence and also to show <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(r\)</EquationSource> <!--MMStat2670003Ferger-m7--> </InlineEquation>-qick convergence in the sense of Strasser.</p>

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Supremal Inequalities for Convex M-Estimators with Applications to Complete and Quick Convergence

  • 期刊论文

  • 发表日期：

    2026年3月1日

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  • Dietmar Ferger

摘要

Abstract

Let \(\hat{m}_{n}\) be an \(M\) -estimator for a parameter \(m\) based on a sample of size \(n\in\mathbb{N}\) . We derive exponential and polynomial upper bounds for the tail-probabilities of \(\sup_{k\geq n}|\hat{m}_{k}-m|\) according as a boundedness- or a moment-condition is fulfilled. This enables us to derive rates of \(r\) -complete convergence and also to show \(r\) -qick convergence in the sense of Strasser.