The evaluation of evidence for exponentially distributed data

At present, likelihood ratios for two-level models are determined with the use of a normal kernel estimation procedure when the between-group distribution is thought to be non-normal. An extension is described here for a two-level model in which the between-group distribution is very positively skewed and an exponential distribution may be thought to represent a good model. The theoretical likelihood ratio is derived. A likelihood ratio based on a biweight kernel with an adaptation at the boundary is developed. The performance of this kernel is compared alongside those of normal kernels and normal and exponential parametric models. A comparison of performance is made for simulated data where results may be compared with those of theory, using the theoretical model, as the true parameter values for the models are known. There is also a comparison for forensic data, using the concentration of aluminium in glass as an exemplar. Performance is assessed by determining the numbers of occasions on which the likelihood ratios for sets of fragments from the same group are supportive of the proposition that they are from different groups and the numbers of occasions on which the likelihood ratios for sets of fragments from different group are supportive of the proposition that they are from the same group.