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Uniform strong consistency of sample quantiles

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  • Zielinski, Ryszard

Abstract

It is well known that if xq(F) is the unique qth quantile of a distribution function F, then Xk(n):n with k(n)/n --> q is a strongly consistent estimator of xq(F). However, for every [var epsilon] >0 and for every, even very large n, supF[set membership, variant]F,PF{Xk(n):n--Xq(F)>[var epsilon]}=1. This is a consequence of the fact that in the family of all distribution functions with uniquely defined qth quantile the almost sure convergence Xk(n):n --> xq(F) is not uniform. A simple necessary and sufficient condition for the uniform strong consistency of Xk(n):n is given.

Suggested Citation

  • Zielinski, Ryszard, 1998. "Uniform strong consistency of sample quantiles," Statistics & Probability Letters, Elsevier, vol. 37(2), pages 115-119, February.
    • Handle: RePEc:eee:stapro:v:37:y:1998:i:2:p:115-119
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    Cited by:

    1. Okolewski, Andrzej & Rychlik, Tomasz, 2001. "Sharp distribution-free bounds on the bias in estimating quantiles via order statistics," Statistics & Probability Letters, Elsevier, vol. 52(2), pages 207-213, April.
    2. Berrendero, José R., 2003. "Uniform strong consistency of robust estimators," Statistics & Probability Letters, Elsevier, vol. 64(2), pages 159-168, August.

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