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

In statistical analysis, we usually use the observed sample values x1, . . . , xn to compute the values of several statistics v(x1, . . . , xn) – such as sample mean, sample variance, etc. The usual formulas for these statistics implicitly assume that we know the exact values x1, . . . , xn. In practice, the sample values x̃1, . . . , x̃n come from measurements and are, thus, only approximations to the actual (unknown) values x1, . . . , xn of the corresponding quantity. Often, the only information that we have about each measurement error ∆xi def = x̃i−xi is the upper bound ∆i on the measurement error: |∆xi| ≤ ∆i. In this case, the only information about each actual value xi is that it belongs to the interval [x̃i−∆i, x̃i+∆i]. It is therefore desirable to compute the range of each given statistic v(x1, . . . , xn) 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. 1 Formulation of the Problem Statistics: reminder. In statistical analysis, we often need to compute some values based on the given sample x1, . . . , xn. For example, usually, we compute the sample mean μ = 1 n · n ∑ i=1 xi and the sample variance σ 2 = 1 n− 1 · n ∑ i=1 (xi−μ). The sample mean is a good approximation to the mean of the corresponding probability distribution, and the sample variance is a good approximation to the variance of this distribution. Alternatively, we can estimate other approximations to mean and variance or approximations to other characteristics of the probability distribution. In all these cases, we compute some value v(x1, . . . , xn) that depends on the sample