Statistical Methods for Investigating the Cosmic Ray Energy Spectrum

Two separate statistical tests are described and developed in order to test unbinned data sets for adherence to the power-law form. The first test employs the TP-statistic, a function defined to deviate from zero when the sample deviates from the power-law form, regardless of the value of the power index. The second test employs a likelihood ratio test to reject a power-law background in favor of a model signal distribution with a cut-off. Introduction and Formalism The question of whether the cosmic ray energy spectrum exhibits a cut-off at the very highest energies is of central interest to the cosmic ray (CR) physics[4, 10]. The flux of CR’s at these energies is very small about 3/km steradian century and, therefore, statistical analysis techniques which clearly quantify ones knowledge of flux suppression are useful. In this note we apply the statistics first developed for binned CR data sets in [6] to an un-binned analysis. We also introduce a new test based on a likelihood ratio test and show that both statistics can quantify our knowledge of a flux suppression. We first establish the mathematical foundations of the analysis. The CR flux follows a power-law for over 10 orders of magnitude. The fundamental probability distribution function (p.d.f.) governing the powerlaw assumption (normalized such that 〈〉X ≡ ∫∞ xmin fX(x;xmin, γ)dx = 1) is fX(x;xmin, γ) = Ax −γ , (1) where A = (γ− 1)x min and the parameter γ is referred to as the spectral index. The n raw moment of this distribution diverges[8] for n ≥ 2 with γ ≤ 3. Alternatively, the expected value of ln(x/xmin) is better behaved and offers a crucial result of this analysis. Analytically we find,