Consider a discrete set of objects and a sample of size N taken with replacement from the set, producing a list of counts of the objects that corresponds to a partition of N. Two statistics that are commonly used for measuring the "diversity" of the sample are the Gini-Simpson index and the Shannon index. We study the number of possible values that these indices can take across all possible partitions of the sample size N as N increases. The two statistics are highly correlated over the set of partitions of N. However, the number of possible values that the Shannon index can take (A383683) far exceeds the number of possible values of the Gini-Simpson index (A069999), with the latter growing quadratically and the former growing faster than every polynomial.
分享
引用
