Uncertainty analysis by moments for asymmetric variables

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.