Processing Quantities with Heavy-Tailed Distribution of Measurement Uncertainty: How to Estimate the Tails of the Results of Data Processing

Measurements are never absolutely accurate; so, it is important to estimate how the measurement uncertainty affects the result of data processing. Traditionally, this problem is solved under the assumption that the probability distributions of measurement errors are normal – or at least are concentrated, with high certainty, on a reasonably small interval. In practice, the distribution of measurement errors is sometimes heavy-tailed, when very large values have a reasonable probability. In this paper, we analyze the corresponding problem of estimating the tail of the result of data processing in such situations. 1 Formulation of the Problem Need for data processing. In many practical situations, we are interested in the values of a quantity y which is not easy (or even impossible) to measure directly: for example, we may be interested in tomorrow’s weather, in the distance to a faraway planet, in the amount of oil in an oil well, etc. In such situations in which we cannot measure y directly, we can often measure y indirectly, i.e.: – measure the values of auxiliary quantities x1, . . . , xn which are related to the desired quantity y by a known relation y = f(x1, . . . , xn), and then – use the results x̃1, . . . , x̃n of measuring the quantities xi and the known dependence to compute the estimate ỹ = f(x̃1, . . . , x̃n) for y. The process of computing ỹ = f(x̃1, . . . , x̃n) is known as data processing. Need to estimating uncertainty of the result of data processing. Measurements are never 100% accurate; so, in general, the measurement results x̃i are somewhat different from the actual values xi of the corresponding quantities. Because of these measurement errors, the estimate ỹ = f(x̃1, . . . , x̃n) is, in general, different from the desired value y = f(x1, . . . , xn) (often, there is an additional difference cause by the fact that the dependence between y and xi is only approximately known). It is therefore important not just to generate an estimate ỹ, but also to gauge how much the actual value y can differ from this estimate, i.e., what is the uncertainty of the result of data processing; see, e.g., [7]. Estimating uncertainty of the result of data processing: traditional statistical approach. Usually, there are many different (and independent) factors which contribute to the measurement error. In many such situations, it is possible to apply the Central Limit Theorem (see, e.g., [9]), according to which, under reasonable conditions, the distribution of the joint effect of numerous independent factors is close to normal. In such situations, it is therefore reasonable to assume that all the measurement errors ∆xi def = x̃i − xi are independent and normally distributed. To describe a normal distribution, it is sufficient to know the mean μ and the standard deviation σ. Thus, under the normality assumption, to gauge the distribution of each measurement error ∆xi, we must know the mean μi and the standard deviation σi of this measurement error. If the known mean is different from 0, this means that this measuring instrument has a bias; we can always compensate for this bias by subtracting the value μi from all the measured values. After this subtraction, the mean error will become 0. Thus, without losing generality, we can assume that each measurement error is normally distributed with mean 0 and known standard deviation σi. The traditional way of estimating the resulting uncertainty ∆y def = ỹ− y in y is based on this assumption. Specifically, since the measurement errors ∆xi are usually relatively small, we can expand the expression ∆y = ỹ − y = f(x̃1, . . . , x̃n)− f(x1, . . . , xn) = f(x̃1, . . . , x̃n)− f(x̃1 −∆x1, . . . , x̃n −∆xn) in Taylor series in ∆xi, ignore quadratic and higher order terms, and keep only terms in ∆xi in this dependence. As a result, we get an expression