Large standard deviations and logarithmic-normality

Six Standard Deviations Suffice

With χ : Ω → {−1,+1} random, A ∈ A, χ(A) has zero mean and standard deviation at most √ n. If |χ(A)| > 6√n then χ(A) is at least six standard deviations off the mean. The probability of this occurring is very small but a fixed positive constant and the number of sets A is going to infinity. In fact, a random χ almost always will not work. The specific constant 6 (actually 5.32) was the result o...

Estimating true standard deviations

A basic statistic that students learn about in their classes is the standard deviation. Like any statistic, standard deviations are influenced by systematic factors and randomness. I propose that researchers should report “corrected” or “true” standard deviations and I show how to calculate them. The notion of “true” statistics comes out of classical test theory (see Lord and Novick, 1968; Gull...

Six Standard Deviations Suffice

Given n sets on n elements it is shown that there exists a two-coloring such that all sets have discrepancy at most Kn>/2, K an absolute constant. This improves the basic probabilistic method with which K = c(ln«)1/2. The result is extended to 11 finite sets of arbitrary size. Probabilistic techniques are melded with the pigeonhole principle. An alternate proof of the existence of Rudin-Shapiro...

Large Deviations

Large Deviations

This paper is based on Wald Lectures given at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations. 1. Large deviations for sums. The role of " large deviations " is best understood through an example. Suppose that X random variables, for instance, normally distributed with mean zero and variance 1. Then, E e θ(X 1 +···+X n) = E[e θX ...