Interval Estimation of the Herfindahl-Hirschman Index under Incomplete Market Information

An interval estimate is provided for the Herfindahl-Hirschman Index (HHI) when the knowledge about the market is incomplete, and we know just the largest market shares. Two rigorous bounds are provided for the HHI, without any further assumptions. Though the interval gets wider as the sum of the known market shares gets smaller, the estimate proves to be quite tight even when the fraction of the market that we do not know in detail is as high as 30%. This robustness is shown through three examples, considering respectively a set of real data and two sets of synthetic data, with the company sizes (a proxy for market shares) following respectively a Zipf law and a Pareto distribution.