In many real-life situations, it is very difficult or even impossible to directly measure the quantity y in which we are interested: e.g., we cannot directly measure a distance to a distant galaxy or the amount of oil in a given well. Since we cannot measure such quantities directly, we can measure them indirectly: by first measuring some relating quantities x1,...,xn, and then by using the known relation between xi and y to reconstruct the value of the desired quantity y.
In practice, it is often very important to estimate the error of the resulting indirect measurement. In this paper, we describe and compare different deterministic and randomized algorithms for solving this problem in the situation when a program for transforming the estimates X1,...,Xn for xi into an estimate for y is only available as a black box (with no source code at hand).
We consider this problem in two settings: statistical, when measurements errors di=Xi-xi are independent Gaussian random variables with 0 average and known standard deviations Di, and interval, when the only known information about di is that |di| cannot exceed a known bound Di. In statistical setting, we describe the optimal error estimation algorithm; in interval setting, we describe a new algorithm which may be not optimal but which is better than the previously known ones.