#### Publication Date

7-2011

#### Abstract

Traditional statistical estimates C(x_{1}, ..., x_{n}) for different statistical characteristics (such as mean, variance, etc.) implicitly assume that we know the sample values x_{1}, ..., x_{n} exactly. In practice, the sample values X_{i} come from measurements and are, therefore, in general, different from the actual (unknown) values X_{i} of the corresponding quantities. Sometimes, we know the probabilities of different values of the measurement error ΔX_{i} = X_{i} - x_{i}, but often, the only information that we have about the measurement error is the upper bound Δ_{i} 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 x_{i} is that they belong to the intervals [X_{i} - Δ_{i}, X_{i} + Δ_{i}].

In general, different values x_{i} from the corresponding interval [X_{i} - Δ_{i}, X_{i} + Δ_{i}] lead to different values of the corresponding statistical characteristic C(x_{1}, ..., x_{n}). In this case, it is desirable to find the set of all possible values of this characteristic. For continuous estimates C(x_{1}, ..., x_{n}), 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.

## Comments

Technical Report: UTEP-CS-11-39