Estimating Statistical Characteristics of Lognormal and Delta-Lognormal Distributions under Interval Uncertainty: Algorithms and Computational Complexity

Traditional statistical estimates Ŝ(x1, . . . , xn) for different statistical characteristics S (such as mean, variance, etc.) implicitly assume that we know the sample values x1, . . . , xn exactly. In practice, the sample values x̃i come from measurements and are, therefore, in general, different from the actual (unknown) values xi of the corresponding quantities. Sometimes, we know the probabilities of different values of the measurement error ∆xi = x̃i − xi, but often, the only information that we have about the measurement error is the upper bound ∆i on its absolute value – provided by the manufacturer of the corresponding measuring instrument. In this case, the only information that we have about the actual values xi is that they belong to the intervals [x̃i −∆i, x̃i +∆i]. In general, different values xi ∈ [x̃i−∆i, x̃i+∆i] lead to different values of the corresponding estimate Ŝ(x1, . . . , xn). In this case, it is desirable to find the range of all possible values of this characteristic. In this paper, we consider the problem of computing the corresponding range for the cases of lognormal and delta-lognormal distributions. Interestingly, it turns out that, in contrast to the case of normal distribution for which it is feasible to compute the range of the mean, for lognormal and delta-lognormal distributions, computing the range of the mean is an NP-hard problem.