Remarks on the L1 distance in statistical data analysis

ABSTRACTWe propose the L1 distance between the distribution of a binned data sample and a probability distribution from which it is hypothetically drawn as a statistic for testing agreement between the data and a model. We study the distribution of this distance for N-element samples drawn from k bins of equal probability and derive asymptotic formulae for the mean and dispersion of L1 in the large-N limit. We argue that the L1 distance is asymptotically normally distributed, with the mean and dispersion being accurately reproduced by asymptotic formulae even for moderately large values of N and k.