Given $(n_{11}, n_{12},n_{21},n_{22}) \sim \text{Mult }(n; \pi_{11}, \pi_{12},\pi_{21},\pi_{22})$, consider the problem of testing the null hypothesis $H_0: \pi_{11} \pi_{22}=\pi_{12}\pi_{21} $ against its one- or two-sided alternative.
a. Formulate the underlying testing problem as that of testing one of the parameters in a multi-parameter exponential family, having expressed the family explicitly in terms of all the parameters and the corresponding statistics.
b. Identify the conditional distribution that is needed to construct the UMPU test?
My attempt is to e express $ \pi_{11} \pi_{22}=\pi_{12}\pi_{21} $ , I can use the ratio $\frac{a+d}{b+c}$, where $a,b,c,d$ are the observed values of $n_{11}, n_{12},n_{21},n_{22}$ group.