Why the Range of a Robust Statistic Under Interval Uncertainty Is Often Easier to Compute

Authors

Olga Kosheleva, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

11-2015

Comments

Technical Report: UTEP-CS-15-87

To appear in Journal of Innovative Technology and Education

Abstract

In statistical analysis, we usually use the observed sample values x₁, ..., x_n to compute the values of several statistics v(x₁, ..., x_n) -- such as sample mean, sample variance, etc. The usual formulas for these statistics implicitly assume that we know the exact values x₁, ..., x_n. In practice, the sample values X₁, ..., X_n come from measurements and are, thus, only approximations to the actual (unknown) values x₁, ..., x_n of the corresponding quantity. Often, the only information that we have about each measurement error Δ x_i = X_i − x_i is the upper bound Δ_i on the measurement error: |Δ x_i| ≤ Δ_i. In this case, the only information about each actual value x_i is that it belongs to the interval [X_i − Δ_i, X_i + Δ_i]. It is therefore desirable to compute the range of each given statistic v(x₁, ..., x_n) over these intervals. It is known that often, estimating the range of a robust statistic (e.g., median) is computationally easier than estimating the range of its traditional equivalent (e.g., mean). In this paper, we provide a qualitative explanation for this phenomenon.