Estimating sample mean under interval uncertainty and constraint on sample variance

摘要

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 i-th 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.