Triangular approximations for continuous random variables in risk analysis
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Triangular approximations for continuous random variables in risk analysis D Johnson* Loughborough University, Leicestershire, UK This paper examines further the problem of approximating the distribution of a continuous random variable based on three key percentiles, typically the median (50th percentile) and the 5% points (5th and 95th percentiles). This usually involves the two main distribution parameters, the mean and standard deviation, and, if possible, the distribution function concerned. Previous research has shown that the Pearson–Tukey formulae provide highly accurate estimates of the mean and standard deviation of a beta distribution (of the first kind), and that simple modifications to the standard deviation formula will improve the accuracy even further. However, little work has been done to establish the accuracy of these formulae for other distributions, or to examine the accuracy of alternative formulae based on triangular distribution approximations. We show that the Pearson–Tukey mean approximation remains highly accurate for a range of unbounded distributions, although the accuracy in these cases can be improved by a slightly different 3:10:3 weighting of the 5%, 50% and 95% points. In contrast, the Pearson–Tukey standard deviation formula is much less accurate for unbounded distributions, and can be bettered by a triangular approximation whose parameters are estimated from simple linear combinations of the three percentile points. In addition, triangular approximations allow the underlying distribution function to be estimated by a triangular cdf. It is shown that simple formulae for estimating the triangular parameters, involving weights of 23:6:1, 13:42:13 and 1:6:23, give not only universally accurate mean and standard deviation estimates, but also provide a good fit to the distribution function with a Kolmogorov–Smirov statistic which averages 0.1 across a wide range of distributions, and an even better fit for distributions which are not highly skewed. Journal of the Operational Research Society (2002) 53, 457–467. DOI: 10.1057=palgrave=jors=2601330 Keywords: risk analysis; estimation; triangular distribution; beta distribution
Introduction Probably the most basic requirement in risk analysis is to estimate reliably the ‘profile’ of any uncertain quantity in terms of an appropriate cumulative distribution function (cdf ) and=or its parameters, particularly the mean and standard deviation. Most decision-makers are likely to find it difficult to estimate these parameters directly, particularly if the distribution is skewed where the effect of the skewness will often be difficult to assess. Likewise, few managers would naturally think of uncertainty as a probability distribution and so could not be expected to reliably assess the shape of a density or distribution function. For most managers, the language of uncertainty is in terms of likelihood associated with simple compariso
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