Fitting the Negative Binomial Distribution

This note is a reaction to recent papers in this journal by Willson, Folks, and Young (1984) and Bowman (1984), which restrict attention to those random variables which are restricted to the lattice of nonnegative integers.

Abstract

This note is a reaction to recent papers in this journal by Willson, Folks, and Young (1984) and Bowman (1984). For the biometrical analysis of certain kinds of observations, such as insect counts, accident counts, or cave entrance counts, when only nonnegative integers are observable, it is expedient to restrict attention to those random variables which are restricted to the lattice of nonnegative integers (cf. Noack, 1950). With an admitted loss of generality, one may substantially simplify the procedure by a further restriction: using as a frame of reference the family of binomial frequency functions (Hoel, 1947). Within this frame of reference, the use of the modem restatement of the time-honored Occam's Razor principle (Wrinch and Jeffreys, 1921; quoted by Popper, 1959, p. 171) leads rather straightforwardly to a null hypothesis, which may be discussed expediently in terms of the number N of distinct, but not necessarily all unequal, observations rj of R (j = 1, . . ., N). Writing