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Order Statistics and Shifted Almost Benford Behavior, preprint

By Steven J. Miller, Mark and J. Nigrini


Abstract. Fix a base B and let ζ have the standard exponential distribution; the distribution of digits of ζ base B is known to be very close to Benford’s Law. If there exists a C such that the distribution of digits of C times the elements of the system is the same as that of ζ, we say the system exhibits Shifted Almost Benford behavior base B (with a shift of log B C mod 1). Let X1,..., XN be independent identically distributed random variables. If the Xi’s are drawn from the uniform distribution [0, L], then as N → ∞ the distribution of the digits of the differences between adjacent Xi’s converges to Shifted Almost Benford behavior (with a shift of log B L/N). Fix a δ ∈ (0, 1) and choose N independent random variables from a nice probability density. The distribution of digits of any N δ consecutive differences and all N − 1 normalized differences of the Xi’s exhibit Shifted Almost Benford behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the un-normalized differences converges to Benford’s Law, Shifted Almost Benford behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior. 1

Year: 2012
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