This paper deals with the empirical Bayes two-action problem of testing $H_0;:;{ heta}{leq}{ heta}_0$: versus $H_1;:;{ heta}>{ heta}_0$ using a linear error loss for some discrete nonexponential families having probability function either $$f_1(x{mid}{ heta})=(x{alpha}+1-{ heta}){ heta}^xprodlimits_{j=0}^x;(j{alpha}+1)$$ or $$f_2(x{mid}{ heta})=[{ heta}prodlimits_{j=0}^{x-1}(j{alpha}+1-{ heta})]/[prodlimits_{j=0}^x;(j{alpha}+1)]$$. Two empirical Bayes tests ${delta}_n^*;and;{delta}_n^{**}$ are constructed. We have shown that both ${delta}_n^*;and;{delta}_n^{**}$ are asymptotically optimal, and their regrets converge to zero at an exponential decay rate O(exp(-cn)) for some c>0, where n is the number of historical data available when the present decision problem is considered.