Fisher information and Cramér-Rao bound for unknown systematic errors

In order to understand the lower bound of achievable measurement uncertainties, the Cramér-Rao inequality is known to be an utmost useful tool. However, the calculation of the Cramér-Rao bound requires a known probability density function that describes the occurring stochastic process. For this reason, the Cramér-Rao bound is applied for determining the lower limit of the measurement uncertainty due to random errors. According to the international guide to the expression of uncertainty in measurement (GUM), unknown systematic errors shall be treated as random errors. This approach is adopted here to enhance the applicability of the Cramér-Rao bound for unknown systematic errors. As a key result, the concept of Fisher information and the Cramér-Rao bound is shown to be applicable also to unknown systematic errors, which is demonstrated for several examples. An unknown offset, an unknown linear drift and successive unknown linear drifts are investigated in detail as systematic errors. Each derived corresponding Fisher information shows a characteristic behavior with respect to the measurement time. In contrast to random errors with a constant variance, the Fisher information can decrease for unknown systematic errors and, thus, the Cramér-Rao bound can increase with an increasing measurement time. For the typically existing case of simultaneously occurring random and unknown systematic errors, an optimal measurement time exists for which the achievable measurement uncertainty becomes minimal. In summary, the examples demonstrate how to determine the Fisher information and the Cramér-Rao bound for unknown systematic errors.