Maximum Likelihood Approach to Pointwise Estimation in Statistical Data Processing under Interval Uncertainty

Authors

Nitaya Buntao, King Mongkut's Institute of Technology North Bangkok
Sa-aat Niwitpong, King Mongkut's Institute of Technology North Bangkok
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

7-2011

Comments

Technical Report: UTEP-CS-11-39

Abstract

Traditional statistical estimates C(x₁, ..., x_n) for different statistical characteristics (such as mean, variance, etc.) implicitly assume that we know the sample values x₁, ..., 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(x₁, ..., x_n). In this case, it is desirable to find the set of all possible values of this characteristic. For continuous estimates C(x₁, ..., 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.