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A uniform L1 law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator

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  • Lafaye de Micheaux, Pierre
  • Ouimet, Frédéric

Abstract

We develop a new L1 law of large numbers where the ith summand is given by a function h(⋅) evaluated at Xi−θn, and where θn≗θn(X1,X2,…,Xn) is an estimator converging in probability to some parameter θ∈R. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating between θ and another consistent estimator θn⋆. Our main contribution is the treatment of the case where |h| blows up at 0, which is not covered by standard uniform laws of large numbers.

Suggested Citation

  • Lafaye de Micheaux, Pierre & Ouimet, Frédéric, 2018. "A uniform L1 law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 109-117.
    • Handle: RePEc:eee:stapro:v:142:y:2018:i:c:p:109-117
    DOI: 10.1016/j.spl.2018.06.006
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    References listed on IDEAS

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    1. Ivana Komunjer, 2007. "Asymmetric power distribution: Theory and applications to risk measurement," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(5), pages 891-921.
    2. A. Desgagné & P. Lafaye de Micheaux, 2018. "A powerful and interpretable alternative to the Jarque–Bera test of normality based on 2nd-power skewness and kurtosis, using the Rao's score test on the APD family," Journal of Applied Statistics, Taylor & Francis Journals, vol. 45(13), pages 2307-2327, October.
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