Abstract
The generalized negative binomial distribution (GNB) was defined by Jamn and Consul (SIAM J. Appl. Math., 21 (1971)), and was obtained as a particular family of Lagrangian-distributions by Consul and Shenton (SIAM J. Appl. Math., 23 (1972)). It has been shown by the conditions of Lagrange theorem that the GNB is a true probability distribution only for $b = 0$ and for all values of b in $p\leqq pb\leqq 1$ and not for any other values. The moments do not exist when $b = p^{ - 1} $. A new statistic T, based upon the sample values $X_{1},X_{2}, \cdots ,X_N $ is also defined which has the property of zero regression on the statistic $\Lambda = X_1 + \cdots + X_N $. This property of zero regression between the two statistics characterizes the GNB.
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