Estimating sample mean under interval uncertainty and constraint on sample variance

Abstract

Traditionally, practitioners start a statistical analysis of a given sample x 1, … , x n by computing the sample mean E and the sample variance V. The sample values x i usually come from measurements. Measurements are never absolutely accurate and often, the only information that we have about the corresponding measurement errors are the upper bounds Δ i on these errors. In such situations, after obtaining the measurement result x ˜ i , the only information that we have about the actual (unknown) value x i of the ith quantity is that x i belongs to the interval x i = [ x ˜ i - Δ i , x ˜ i + Δ i ] . Different values x i from the corresponding intervals lead, in general, to different values of the sample mean and sample variance. It is therefore desirable to find the range of possible values of these characteristics when x i ∈ x i . Often, we know that the values x i cannot differ too much from each other, i.e., we know the upper bound V 0 on the sample variance V : V ⩽ V 0. It is therefore desirable to find the range of E under this constraint. This is the main problem that we solve in this paper.