Monotone empirical Bayes tests for some discrete nonexponential families

This paper deals with the empirical Bayes two-action problem of testingH ₀ : θ ≤ θ_o versusH ₁ : θ > θ_o using a linear error loss for some discrete nonexponential families having probability function either $\begin{gathered} f_1 (x|\theta ) = (x\alpha + 1 - \theta )\theta ^x /\prod\limits_{j = 0}^x {(j\alpha + 1)} \\ or \\ f_1 (x|\theta ) = \left {\theta \prod\limits_{j = 0}^{x - 1} {(j\alpha + 1 - \theta )} } \right]/\left {\prod\limits_{j = 0}^x {(j\alpha + 1)} } \right] \\ \end{gathered} $ Two empirical Bayes tests δ_n* and δ_n** are constructed. We have shown that both δ_n* and δ_n** are asymptotically optimal, and their regrets converge to zero at an exponential decay rate O(exp( -cn)) for some c > 0, wheren is the number of historical data available when the present decision problem is considered.