Statistical properties of maximum likelihood estimators of power law spectra information

A maximum likelihood (ML) procedure was developed (Nucl. Instr. and Meth. Phys. Res. 480 (2-3) (2002) 739) for estimating the single parameter of a simple power law energy spectrum and generalized (Nucl. Instr. and Meth. Phys. Res. A 489 (2002) 422) to estimate the three spectral parameters of the broken power law energy spectrum from simulated detector responses and real cosmic-ray data. The statistical properties of the ML estimator are investigated in this paper and shown to have the three desirable properties: (P1) consistency (unbiased), (P2) efficiency (attains the Cramer-Rao minimum variance bound), and (P3) normally distributed, under a wide range of potential detector response functions. While simulation studies can easily determine if a given estimation procedure provides unbiased spectra information and whether or not the estimator is approximately normally distributed, attainment of the Cramer-Rao bound (CRB) can only be determined by calculating the CRB for an assumed energy spectrum-detector response function combination, which can be quite formidable in practice. However, the effort in calculating the CRB is very worthwhile because it provides: (1) the fundamental limit to the precision of spectral parameter determination for a cosmic-ray experiment, (2) the necessary means to compare the efficiency of competing estimation techniques and, (3) a stopping rule in the search for the best unbiased estimator. Consequently, the CRB for both the simple and broken power law energy spectra are derived herein and the conditions under which they are attained in practice are investigated. The impact of systematic errors in a detector on the efforts to determine the spectra information is also investigated