Order Statistics and Benford's Law

Fix a base B and let zeta have the standard exponential distribution; the distribution of digits of zeta 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 some set is the same as that of zeta, we say that set exhibits shifted exponential behavior base B (with a shift of log_B C \bmod 1). Let X_1, >..., X_N be independent identically distributed random variables. If the X_i's are drawn from the uniform distribution on [0,L], then as N\to\infty the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior (with a shift of \log_B L/N \bmod 1). By differentiating the cumulative distribution function of the logarithms modulo 1, applying Poisson Summation and then integrating the resulting expression, we derive rapidly converging explicit formulas measuring the deviations from Benford's Law. Fix a delta in (0,1) and choose N independent random variables from any compactly supported distribution with uniformly bounded first and second derivatives and a second order Taylor series expansion at each point. The distribution of digits of any N^\delta consecutive differences \emph{and} all N-1 normalized differences of the order statistics exhibit shifted exponential 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 exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.Comment: 14 pages, 2 figures, version 4: Version 3: most of the numerical simulations on shifted exponential behavior have been suppressed (though are available from the authors upon request). Version 4: a referee pointed out that we need epsilon > 1/3 - delta/2 in the proof of Theorem 1.5; this has now been adde

Biases in the reporting of hepatocellular carcinoma tumor sizes on the liver transplant waiting list.

We investigated the possibility that patients with hepatocellular carcinoma (HCC) listed for liver transplant with tumors just outside stage T2 size criteria may be inaccurately reported as just meeting the tumor size criteria for transplant. The United Network for Organ Sharing/Standard Transplant Analysis and Research database identified 12,958 patients listed for liver transplants with HCC exception points from 2006 to 2013, 9,168 of whom were listed with one tumor. A logistic power peak function was fitted to the single-tumor size histogram, with the fitted values representing unbiased expected values. The difference between the observed and expected tumor counts for 2.0 cm and 5.0 cm was 238 (22%) and 66 (57%), respectively. This suggests that up to 304 (3.0%) patients with tumors outside of transplant criteria had their measurements recorded at the margins of eligibility. A risk-adjusted Poisson model evaluated the ratio of observed to expected HCC recurrence by tumor size. There were 435 HCC recurrences among 6,049 transplants. Only 2.0-cm tumors had observed to expected recurrence differing from 1 (ratio 0.73, 95% confidence interval 0.57-0.94), indicating a 27% lower than expected rate of recurrence.ConclusionHigher than expected observed tumor counts at the lower transplant criteria margin were corroborated by lower than expected HCC recurrence, suggesting that tumor sizes at the margins of HCC transplant criteria may be subject to inaccurate reporting. (Hepatology 2017;66:1144-1150)