Evaluating the Maximum Likelihood Method for Detecting Short-Term Variability of AGILE gamma-ray Sources

The AGILE space mission (whose instrument is sensitive in the energy ranges 18-60 keV, and 30 MeV - 50 GeV) has been operating since 2007. Assessing the statistical significance of time variability of gamma-ray sources above 100 MeV is a primary task of the AGILE data analysis. In particular, it is important to check the instrument sensitivity in terms of Poisson modeling of the data background, and to determine the post-trial confidence of detections. The goals of this work are: (i) evaluating the distributions of the likelihood ratio test for "empty" fields, and for regions of the Galactic plane; (ii) calculating the probability of false detection over multiple time intervals. In this paper we describe in detail the techniques used to search for short-term variability in the AGILE gamma-ray source database. We describe the binned maximum likelihood method used for the analysis of AGILE data, and the numerical simulations that support the characterization of the statistical analysis. We apply our method to both Galactic and extra-galactic transients, and provide a few examples. After having checked the reliability of the statistical description tested with the real AGILE data, we obtain the distribution of p-values for blind and specific source searches. We apply our results to the determination of the post-trial statistical significance of detections of transient gamma-ray sources in terms of pre-trial values. The results of our analysis allow a precise determination of the post-trial significance of {\gamma}-ray sources detected by AGILE.