Maximum likelihood estimation of spectra information from multiple independent cosmic ray data sets

Abstract

A procedure based on the method of maximum likelihood (ML) is developed to allow an arbitrary number of cosmic ray data sets produced by different instruments with different energy ranges and resolutions to be used together in the analysis of the spectra. Application of this approach will facilitate the interpretation of energy spectra data from multiple science missions and thereby provide more accurate spectral parameter estimates based on the combination of data sets. The benefits of this technique are measured in terms of the reduction of the statistical errors (standard deviations) and biases of the spectra information using the multiple data sets in concert as compared to the statistical errors of the spectra information when the data sets are considered separately. The Cramer–Rao bound (CRB) is derived for multiple independent data sets and provides the fundamental limit to the precision of spectral parameter determination from the combination of cosmic ray data sets. The CRB is used to quantify the efficiency of the derived procedure and also provides a stopping rule in the search for the best unbiased methodology for quantifying the parameters from the data sets. Application of the ML procedure is demonstrated on two simulated data sets acting singly and then together in the estimation of the three broken power law spectral parameters, along with the numerical details required for its successful implementation in practice. Examples of several detector systematic errors and their impact on the efforts to measure the spectra are also investigated.