A note on error estimation for hypothesis testing problems for some linear SPDEs

In the recent paper [CX13], we study the simple hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive space-time white noise colored in space. Assuming that one path of the projected solution is observed continuously over time interval $[0,T]$, we established `the proper asymptotic classes' of tests in which we can find `asymptotically the most powerful test' -- tests with fastest speed of error convergence -- in two asymptotic regimes: large time asymptotics $T\to\infty$, and increasing number of Fourier modes $N\to\infty$. The main goal of the present paper is to study how to estimate and control the Type I and Type II errors for finite $T$ and $N$, in a feasible and practical way. We propose a new Likelihood Ratio type rejection regions and derive explicit expressions for thresholds for $T$, and $N$ that will guarantee that the corresponding statistical errors are smaller than a given upper bound. The key ideas of the proofs are essentially based on some results on sharp large deviation principles developed in [CX13]. The theoretical results are illustrated by means of numerical simulations.