Reducing Bias in Estimates for the Law of Crime Concentration

ObjectivesThe law of crime concentration states that half of the cumulative crime in a city will occur within approximately 4% of the city's geography. The law is demonstrated by counting the number of incidents in each of N spatial areas (street segments or grid cells) and then computing a parameter based on the counts, such as a point estimate on the Lorenz curve or the Gini index. Here we show that estimators commonly used in the literature for these statistics are biased when the number of incidents is low (several thousand or less). Our objective is to significantly reduce bias in estimators for the law of crime concentration.MethodsBy modeling crime counts as a negative binomial, we show how to compute an improved estimate of the law of crime concentration at low event counts that significantly reduces bias. In particular, we use the Poisson-Gamma representation of the negative binomial and compute the concentration statistic via integrals for the Lorenz curve and Gini index of the inferred continuous Gamma distribution.ResultsWe illustrate the Poisson-Gamma method with synthetic data along with homicide data from Chicago. We show that our estimator significantly reduces bias and is able to recover the true law of crime concentration with only several hundred events.ConclusionsThe Poisson-Gamma method has applications to measuring the concentration of rare events, comparisons of concentration across cities of different sizes, and improving time series estimates of crime concentration.