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Asymptotic Theory of $L$-Statistics and Integrable Empirical Processes

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  • Tetsuya Kaji

Abstract

This paper develops asymptotic theory of integrals of empirical quantile functions with respect to random weight functions, which is an extension of classical $L$-statistics. They appear when sample trimming or Winsorization is applied to asymptotically linear estimators. The key idea is to consider empirical processes in the spaces appropriate for integration. First, we characterize weak convergence of empirical distribution functions and random weight functions in the space of bounded integrable functions. Second, we establish the delta method for empirical quantile functions as integrable functions. Third, we derive the delta method for $L$-statistics. Finally, we prove weak convergence of their bootstrap processes, showing validity of nonparametric bootstrap.

Suggested Citation

  • Tetsuya Kaji, 2019. "Asymptotic Theory of $L$-Statistics and Integrable Empirical Processes," Papers 1910.07572, arXiv.org.
    • Handle: RePEc:arx:papers:1910.07572
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    Cited by:

    1. Hongyi Jiang & Zhenting Sun & Shiyun Hu, 2023. "A Nonparametric Test of $m$th-degree Inverse Stochastic Dominance," Papers 2306.12271, arXiv.org, revised Jul 2023.
    2. Xiaojun Song & Zhenting Sun, 2023. "Almost Dominance: Inference and Application," Papers 2312.02288, arXiv.org, revised Oct 2025.
    3. Firpo, Sergio & Galvao, Antonio F. & Parker, Thomas, 2023. "Uniform inference for value functions," Journal of Econometrics, Elsevier, vol. 235(2), pages 1680-1699.
    4. Amparo Ba'illo & Javier C'arcamo & Carlos Mora-Corral, 2021. "Extremal points of Lorenz curves and applications to inequality analysis," Papers 2103.03286, arXiv.org.

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