Traditional statistical estimates C(x1, ..., xn) for different statistical characteristics (such as mean, variance, etc.) implicitly assume that we know the sample values x1, ..., xn exactly. In practice, the sample values Xi come from measurements and are, therefore, in general, different from the actual (unknown) values Xi of the corresponding quantities. Sometimes, we know the probabilities of different values of the measurement error ΔXi = Xi - xi, but often, the only information that we have about the measurement error is the upper bound Δi on its absolute value -- provided by the manufacturer of the corresponding measuring instrument. In this case, the only information that we have about the actual values xi is that they belong to the intervals [Xi - Δi, Xi + Δi].
In general, different values xi from the corresponding interval [Xi - Δi, Xi + Δi] lead to different values of the corresponding statistical characteristic C(x1, ..., xn). In this case, it is desirable to find the set of all possible values of this characteristic. For continuous estimates C(x1, ..., xn), this range is an interval.
The values of C are used, e.g., in decision making -- e.g., in a control problem, to select an appropriate control value. In this case, we need to select a single value from the corresponding interval. It is reasonable to select a value which is, in some sense, the most probable. In this paper, we show how the Maximum Likelihood approach can provide such a value, i.e., how it can produce pointwise estimates in statistical data processing under interval uncertainty.