Computation of error spectrum for estimation performance evaluation

The error spectrum (ES) of an estimator was proposed as a measure for evaluation of estimation performance. In this paper, we examine the conditions that ES has a positive finite value, which is equivalent to requiring the existence of a finite rth (r is a real number) moment of a non-negative random variable. Different conditions for this existence problem are presented. Further, methods to analytically evaluate ES are elaborated. A clear connection between ES and the Mellin transform provides a means to compute ES analytically. Fractional integration/derivative of the moment generating function of the estimation error can also achieve analytical solutions. ES of the chi distribution, Gamma distribution, Weibull distribution, and Beta distribution are computed to illustrate our methods.