2013
DOI: 10.1214/ecp.v18-2359
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On the concentration of the missing mass

Abstract: A random variable is sampled from a discrete distribution. The missing mass is the probability of the set of points not observed in the sample. We sharpen and simplify McAllester and Ortiz's results (JMLR, 2003) bounding the probability of large deviations of the missing mass. Along the way, we refine and rigorously prove a fundamental inequality of Kearns and Saul (UAI, 1998).

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Cited by 99 publications
(49 citation statements)
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“…where the last line follows directly from Theorem 1.1 in Schlemm (2016) (a result equivalent to Theorem 1.1 was also obtained in Berend and Kontorovich (2013)). Therefore, the standard Hanson-Wright inequality implies that with probability at least 1 − e −t we have…”
Section: Improving Hanson-wright Inequality In the Subgaussian Regimementioning
confidence: 66%
“…where the last line follows directly from Theorem 1.1 in Schlemm (2016) (a result equivalent to Theorem 1.1 was also obtained in Berend and Kontorovich (2013)). Therefore, the standard Hanson-Wright inequality implies that with probability at least 1 − e −t we have…”
Section: Improving Hanson-wright Inequality In the Subgaussian Regimementioning
confidence: 66%
“…thus indicating that 1 2g(µ) is a distribution-sensitive proxy variance for any [0, 1]-supported random variable with mean µ (see also Berend and Kontorovich, 2013, for a detailed proof of this result). If this is the optimal proxy variance for the Bernoulli distribution (see Theorem 2.1 and Theorem 3.1 of Buldygin and Moskvichova, 2013), it is clear from our result that it does not hold true for the Beta distribution.…”
Section: Introductionmentioning
confidence: 79%
“…The sub-Gaussian property Kozachenko, 1980, 2000;Pisier, 2016) and related concentration inequalities (Boucheron et al, 2013;Raginsky and Sason, 2013) have attracted a lot of attention in the last couple of decades due to their applications in various areas such as pure mathematics, physics, information theory and computer sciences. Recent interest focused on deriving the optimal proxy variance for discrete random variables like the Bernoulli distribution (Buldygin and Moskvichova, 2013;Kearns and Saul, 1998;Berend and Kontorovich, 2013) and the missing mass (McAllester and Schapire, 2000;McAllester and Ortiz, 2003;Berend and Kontorovich, 2013;Ben-Hamou et al, 2017). Our focus is instead on two continuous random variables, the Beta and Dirichlet distributions, for which the optimal proxy variance was not known to the best of our knowledge.…”
Section: Introductionmentioning
confidence: 99%
“…Various properties of the Good-Turing estimator and several variations of it have been analyzed for distribution estimation and compression [9], [10], [11], [12], [13], [14], [15]. Several concentration results on missing mass estimation are also known [16], [17]. Despite all this work, the risk of the Good-Turing estimator and the minimax risk of missing mass estimation have still not been conclusively established.…”
Section: A Good-turing Estimator and Previous Resultsmentioning
confidence: 99%