Estimating mean under interval uncertainty and variance constraint

摘要

In many practical situations, we have a sample of objects of a given type. When we measure the values of a certain quantity for these objects, we get a sequence of values x1, …, xn. When the sample is large enough, then the arithmetic mean E of the values xi is a good approximation for the average value of this quantity for all the objects from this class. The values xi come from measurements, and measurements are never absolutely accurate. Often, the only information that we have about the measurement error is the upper bound Δi on this error. In this case, once we have the measurement result xi, the condition that |xi — xi| ≤ Δi implies that the actual (unknown) value xi belongs to the interval [xi — Δi, xi + Δi]. In addition, we often know the upper bound V0 on the variance V of the actual values — e.g., we know that the objects belong to the same species, and we know that within-species differences cannot be too high. In such cases, to estimate the average over the class, we need to find the range of possible values of the mean under the constraints that each xi belongs to the given interval [xi xi] and that the variance V(x1, …, xn) is bounded by a given value V0. In this paper, we provide efficient algorithms for computing this range.