Nemirovski Estimate of Mean of Arbitrary Distributions with Bounded Variance

Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious. 1 Nemirovski Estimate of Mean of Arbitrary Distributions with Bounded Variance The popular Chernoff bounds assume severe restrictions on distribution: it must be cut-off, or vanish exponentially, etc. In [Nemirovsky Yudin 83], an equally simple bound uses no conditions at all beyond independence and known bound on variance. It is not widely used because it is not explained anywhere with very explicit computation. I offer this summary: Assume independent variables xi with the same unknown mean m and known lower bounds bi on inverses 1/vi of variance. We estimate m with < 2 −k chance of error exceeding ε. This requires ∑ bi of about 12k/ε . First, we normalize xi to set ε = 1, spread them into 2k − 1 groups, and in each group j take an average Xj , weighted in proportion to bi. The inverse variance bounds Bj for Xj are additive and we assure Bj ≥ ( √ 2 + 1) = b. By Chebyshev’s inequality, Xj deviate from m to each side by ≥ 1 with probability ≤ 1/(b + 1). (We assume equality: the general case follows by modifying the distribution.) Their median then deviates from m by ≥ 1 with probability P ≤ 2 k−1