Old Bands, New Tracks—Revisiting the Band Model for Robust Hypothesis Testing

Abstract

Abstract-The density band model proposed by Kassam [1] for robust hypothesis testing is revisited. First, a novel criterion for the general characterization of least favorable distributions is proposed that unifies existing results. This criterion is then used to derive an implicit definition of the least favorable distributions under band uncertainties. In contrast to the existing solution, it only requires two scalar values to be determined and eliminates the need for case-by-case statements. Based on this definition, a generic fixed-point algorithm is proposed that iteratively calculates the least favorable distributions for arbitrary band specifications. Finally, three different types of robust tests that emerge from band models are discussed and a numerical example is presented to illustrate their potential use in practice.

Robust distributed sequential hypothesis testing for detecting a random signal in non-Gaussian noise

“…We characterize the set of possible probabilities P i under hypothesis H i using Kassam's band model [8], [9] as…”

On the Minimization of Convex Functionals of Probability Distributions Under Band Constraints

“…Third, from a theoretical point of view, the density band model is of interest because it generalizes several popular uncertainty models such as the ε-contamination model [21] and the bounded distribution function model [22]. A more detailed discussion of the band model and its properties can be found in [6] and [7].…”

“…This model is known as the density band model and is commonly used in robust statistics [5]. In the context of robust hypothesis testing it was first studied in [6] and was recently revisited in [7]. It is discussed in more detail in Section II.…”

On the Equivalence of <tex>$f$</tex>-Divergence Balls and Density Bands in Robust Detection

“…Moreover, the corresponding least favorable densities can be calculated in a generic manner using a simple and efficient algorithm. See [14] for a more detailed discussion of the band model and the calculation of its least favorable densities.…”

“…Using Theorem 2 in [10], the least favorable densities for this model can be calculated efficiently by solving two integral equations. The coefficients for the corresponding band model can be identified by comparing (6) in [10] to (13) and (14) in this paper. For the numerical values given above, they calculate to a0 ≈ 0.9047, b0 ≈ 2.2519, a1 ≈ 0.8319, and b1 ≈ 1.3009.…”