Uncertainty analysis by moments for asymmetric variables

Abstract

This paper gives equations that enable a method of uncertainty analysis recently described (Willink R 2005 Metrologia 42 329–43) to accommodate influence variables with asymmetric probability distributions. The equations permit the calculation of 90%, 95%, 98% and 99% coverage intervals [a, b] for the measurand Y = F(X1,…, Xm) from the first four moments of an approximating distribution. The equations for 90% and 98% are presented so that the one-sided probability statements Pr(Y ⩾ a) = P and Pr(Y ⩽ b) = P can be supplied for the familiar probability levels of P = 95% and P = 99%. Additional equations are provided so that the exact moments can be calculated in certain non-linear problems. An improved method is given for dealing with inputs that have shifted and scaled t-distributions with 4 or fewer degrees of freedom, for which the first four moments do not all exist.

Non-normal distribution of the top-of-atmosphere satellite optical measurements over calibration sites

“…For a non-normal distribution, the area included in the confidence interval [-σ, σ] does not cover the 68.27% (k = 1) of samples. The work initially described in Willink (2005) and refined in Willink (2006) develops a method to calculate the 90%, 95%, 98%, and 99% coverage intervals by using the first four moments of an approximating distribution. Thus, the use of the third and four moments of the distribution (skewness and kurtosis) is a useful mechanism to estimate the variation of the coverage interval independently of changes in the distribution shape in time or space domains.…”

A Model to Determinate the Influence of Probability Density Functions (PDFs) of Input Quantities in Measurements

“…In the case of Hypothesis 1, the probability distribution function F pxq is estimated using a kernel density estimation (KDE) [22,23]; in the case of Hypothesis 2, the probability distribution function F pxq is a known, univariate marginal distribution (normal, t-student, . .…”

Difficulties arising from the representation of the measurand by a probability distribution

“…In many situations it may not be immediately obvious that the distributed-measurand concept is being adopted. For example, this concept is being invoked in a statement like "the probability that the mass of the electron m e exceeds 9.109 382 15 × 10 −31 kg is 0.5" (1) which is a statement that some readers might make because the figure indicated is the 2006 CODATA 4 estimate of m e . If such a statement appears meaningful then the reader is attributing the fundamental constant m e some probability distribution and is knowingly or unknowingly accepting the distributedmeasurand concept.…”

“…The idea of propagating distributions is simply the idea of performing algebra with random variablesirrespective of what these random variables represent. Such algebra, whether it be carried out by exact methods, by the Monte Carlo principle, or by the principle of equating moments [4], is mathematically valid. In summary, our subject relates to the meaning, not the manipulation, of the random variables involved in the evaluation of measurement uncertainty.…”