Range of validity of α and β for a generalized diversity index H(α,β) due to Good

Good (I.J. Good, Biometrika 40 (1953) 237; J. Am. Statist. Assoc. 77 (1982) 561) proposed a diversity index H(α,β), defined for non-negative integer α and β, which generalized the well-known indices of Shannon and Simpson. In this paper we further generalize Goodʹs index by allowing the parameters α and β to take values in the real plane. For two simple properties which a `goodʹ diversity index should possess, we determine the range of valid α and β values for this generalized index.