In search of an unbiased temperature estimator for statistically poor X-ray spectra

Although commonly employed by X-ray astronomers, maximum likelihood estimators are known to be biased. In this paper we investigate the bias associated with the measure of the temperature from an X-ray thermal spectrum. We show that, in the case of low surface brightness regions, commonly adopted estimators, such as those based on χ² and Cash statistics, return strongly biased results. We stress that this can have strong implications when measuring the temperature of cluster outer regions with current experiments. We consider various approaches to overcoming this problem and the most effective is a technique that allows us to correct the bias a posteriori. Extensive Monte Carlo simulations show that our correction returns excellent results under different conditions.