A sharp estimate of the binomial mean absolute deviation with applications

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25 Scopus citations

Abstract

We give simple, sharp non-asymptotic bounds on the mean absolute deviation (MAD) of a Bin (n, p) random variable. Although MAD is known to behave asymptotically as the standard deviation, the convergence is not uniform over the range of p and fails at the endpoints. Our estimates hold for all p ∈ [0, 1] and illustrate a simple transition from the "linear" regime near the endpoints to the "square root" regime elsewhere. As an application, we provide asymptotically optimal tail estimates of the total variation distance between the empirical and the true distributions over countable sets.

Original languageEnglish
Pages (from-to)1254-1259
Number of pages6
JournalStatistics and Probability Letters
Volume83
Issue number4
DOIs
StatePublished - 1 Apr 2013

Keywords

  • Binomial
  • Density estimation
  • Mean absolute deviation
  • Total variation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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