Using the C(α) procedure of Neyman (1959), we derive teats for the goodness of fit of the binomial distribution which are asymptotically optimal against generalized binomial alternatives proposed by Altham (1978) and Kupper & Haseman (1978). The C(α) tests optimal against correlated binomial alternatives, beta-binomial alternatives and a general class of mixture alternatives suggested by WisniewsM (1968) are shown to be closely related to the binomial variance test. A numerical example is given and the small sample behaviour of the variance test and the C(α) tests is investigated in a Monte Carlo sampling experiment. The asymptotic relative efficiencies of the goodness of fit statistics are examined.

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