The Burr family of distributions has been used extensively for over 20 years as a *species sensitivity distribution *(SSD) in ecotoxicology. Indeed, its popularity (at least in Australia and New Zealand) has been guaranteed by the fact that it is a default distribution in the Burrlioz software tool which underpins the methodology for guideline development (GV) in those countries.

The main advantages of the Burr distributions are (i) they can accommodate a wide variety of distributional shapes; and (ii) the *cdf* is readily inverted to provide a closed-form solution for percentile estimates (aka HCx values).

The down-side however, is that many toxicity data sets have characteristics that can result in numerical instabilities when maximum likelihood approaches are used to estimate the parameters of a Burr distribution. The problem is most acute when attempting to fit a 3-parameter Burr distribution to small toxicity data sets that are highly skewed (*statisticians would walk away from such situations, but unfortunately, this is the norm in ecotoxicology*).

We have recently been undertaking research into alternative estimation strategies and/or re-parameterising the Burr distributions to overcome these computational issues. A promising alternative to the use of maximum likelihood is estimation using *L-Moments.*

We have developed a simple, graphical-based method for the L-Moment estimation of the parameters of a Burr III distribution. The video below explains.

So what does this distribution look like and how does it compare to that obtained using MLEs? Figure 1 below says it all – the fits are very similar over the entire range of concentrations and almost identical where it matters – in the left tail and this is reflected in the HCx estimates of Table 1.

Burr III parameters: {b,c,k} | HC1 | HC5 | HC10 |
---|---|---|---|

MLE {6.304,11.932,0.180} | 0.737 | 1.560 | 2.155 |

L-Moments {5.896,8.148,0.273} | 0.744 | 1.534 | 2.095 |