Two approximation algorithms of error spectrum for estimation performance evaluation

Error spectrum is a comprehensive metric for evaluation of estimation performance in that it is an aggregation of many incomprehensive measures. However, error spectrum requires computing the expectation of the rth power of the estimation-error-norm as using it to evaluate an estimator's performance. Therefore unless the error distribution is given, it's usually not easy to obtain the error spectrum. To alleviate this difficulty, two approximation algorithms are proposed. One is the Gaussian mixture method, which calculated the error spectrum by capturing the probability density function. The other using the sample is the power means error method. Furthermore, how the Gaussian mixture method and power means error method can be used in estimation performance evaluation are analyzed not only in the large sample case but also in the small sample case. Numerical examples are provided to illustrate the effectiveness of the above two algorithms. It is shown that the two proposed algorithms can be applied easily to calculate the error spectrum in estimator performance evaluation.