Error Rate for the Limiting Poisson-power Function Distribution

The number of neutron signals from a neutral particle beam(NPB) at the detector, without any errors, obeys Poisson distribution, Under two assumptions that NPB scattering distribution and aiming errors have a circular Gaussian distribution respectively, an exact probability distribution of signals becomes a Poisson-power function distribution. In this paper, we show that the error rate in simple hypothesis testing for the limiting Poisson-power function distribution is not zero. That is, the limit of <TEX>${alpha}+{eta}$</TEX> is zero when Poisson parameter<TEX>$kappa ightarroinfty$</TEX>, but this limit is not zero (i.e., <TEX>$ hoell$</TEX>>0)for the Poisson-power function distribution. We also give optimal decision algorithms for a specified error rate.