A sharp estimate of the binomial mean absolute deviation with applications

Daniel Berend; Aryeh Kontorovich

doi:10.1016/j.spl.2013.01.023

A sharp estimate of the binomial mean absolute deviation with applications

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52 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 language	English
Pages (from-to)	1254-1259
Number of pages	6
Journal	Statistics and Probability Letters
Volume	83
Issue number	4
DOIs	https://doi.org/10.1016/j.spl.2013.01.023
State	Published - 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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10.1016/j.spl.2013.01.023

Cite this

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title = "A sharp estimate of the binomial mean absolute deviation with applications",

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.",

keywords = "Binomial, Density estimation, Mean absolute deviation, Total variation",

author = "Daniel Berend and Aryeh Kontorovich",

note = "Funding Information: We thank Andrew Barron, L{\'a}szl{\'o} Gy{\"o}rfi, John Hartigan, G{\'a}bor Lugosi, David McAllester, Alon Orlitsky, David Pollard, and Larry Wasserman for helpful discussions and comments, and an anonymous referee for corrections. We are grateful to the Stone family for providing a venue for this work. The second author was supported in part by the Israel Science Foundation (grant No. 1141/12 ).",

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AU - Kontorovich, Aryeh

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PY - 2013/4/1

Y1 - 2013/4/1

N2 - 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.

AB - 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.

KW - Binomial

KW - Density estimation

KW - Mean absolute deviation

KW - Total variation

UR - https://www.scopus.com/pages/publications/84874350059

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DO - 10.1016/j.spl.2013.01.023

M3 - Article

AN - SCOPUS:84874350059

SN - 0167-7152

VL - 83

SP - 1254

EP - 1259

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

IS - 4

ER -

A sharp estimate of the binomial mean absolute deviation with applications

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Keywords

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