An unbiased estimate for the mean of a 0,1 random variable with relative error distribution independent of the mean

2013/09/20 by Mark Huber, Huber, Mark
Computer Science · Mathematics · #62F10 #62F25 #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST) #cs.CC #math.PR #math.ST #msc:62F10 #msc:62F25 #stat.TH

paper · pdf · doi:10.48550/arxiv.1309.5413

12 pages; 0 figures

arxiv created 2015/11/17 · arxiv updated 2015/11/18

Abstract

Say X₁,X₂,… are independent identically distributed Bernoulli random variables with mean p. This paper builds a new estimate p of p that has the property that the relative error, p /p - 1, of the estimate does not depend in any way on the value of p. This allows the construction of exact confidence intervals for p of any desired level without needing any sort of limit or approximation. In addition, p is unbiased. For ε and δ in (0,1), to obtain an estimate where ℙ(| p/p - 1| > ε) ≤ δ, the new algorithm takes on average at most 2ε^-2 p^-1ln(2δ^-1)(1 - (14/3) ε)^-1 samples. It is also shown that any such algorithm that applies whenever p ≤ 1/2 requires at least 0.2ε^-2 p^-1ln((2-δ)δ^-1)(1 + 2 ε) samples. The same algorithm can also be applied to estimate the mean of any random variable that falls in [0,1].

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