Taking into Account Interval (and Fuzzy) Uncertainty Can Lead to More Adequate Statistical Estimates

Traditional statistical data processing techniques (such as Least Squares) assume that we know the probability distributions of measurement errors. Often, we do not have full information about these distributions. In some cases, all we know is the bound of the measurement error; in such cases, we can use known interval data processing techniques. Sometimes, this bound is fuzzy; in such cases, we can use known fuzzy data processing techniques. However, in many practical situations, we know the probability distribution of the random component of the measurement error and we know the upper bound – numerical or fuzzy – on the measurement error’s systematic component. For such situations, no general data processing technique is currently known. In this paper, we describe general data processing techniques for such situations, and we show that taking into account interval and fuzzy uncertainty can lead to more adequate statistical estimates. 1 Formulation of the Problem: Traditional Statistical Approach to Data Processing Is Not Always Applicable Data processing: a brief reminder. Some quantities, we can directly measure. For example, on the Earth, we can usually directly measure the distance between the two nearby points. However, many other quantities X j we cannot measure directly. Ligang Sun, Hani Dbouk, Ingo Neumann, Steffen Schön Leibniz Universität Hannover, 30167 Hannover, Germany, e-mail: [email protected], [email protected], [email protected], [email protected] Vladik Kreinovich Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, Texas 79968, USA, e-mail: [email protected]