In this work, based on a realization of an inhomogeneous Poisson process whose intensity function depends on a real parameter, we consider a simple null hypothesis against the composite one sided alternative. Under certain regularity conditions we will obtain the power loss of the score test which measures its performance with respect to the Neyman-Pearson test. We present the second-order approximation of the power of the score test under the close alternatives by specifying the explicit form of the next term after the Gaussian term.
