Cost curves have recently been introduced as an alternative or complement to ROC curves in order to visualize binary classifiers performance. Of importance to both cost and ROC curves is the computation of confidence intervals along with the curves themselves so that the reliability of a classifier's performance can be assessed. Computing confidence intervals for the difference in performance between two classifiers allows the determination of whether one classifier performs significantly better than another. A simple procedure to obtain confidence intervals for costs or the difference between two costs, under various operating conditions, is to perform bootstrap resampling of the test set. In this paper, we derive exact bootstrap distributions for these values and use these dstributions to obtain confidence intervals, under various operating conditions. Performances of these confidence intervals are measured in terms of coverage accuracies. Simulations show excellent results.

Estimating insurance premia from data is a difficult regression problem for several reasons: the large number of variables, many of which are .discrete, and the very peculiar shape of the noise distri(cid:173) bution, asymmetric with fat tails, with a large majority zeros and a few unreliable and very large values. We compare several machine learning methods for estimating insurance premia, and test them on a large data base of car insurance policies. We find that func(cid:173) tion approximation methods that do not optimize a squared loss, like Support Vector Machines regression, do not work well in this context. Compared methods include decision trees and generalized linear models. The best results are obtained with a mixture of experts, which better identifies the least and most risky contracts, and allows to reduce the median premium by charging more to the most risky customers.

Incorporating prior knowledge of a particular task into the architecture of a learning algorithm can greatly improve generalization performance. We study here a case where we know that the function to be learned is non-decreasing in two of its arguments and convex in one of them. For this purpose we propose a class of functions similar to multi-layer neural networks but (1) that has those properties, (2) is a universal approximator of continuous functions with these and other properties. We apply this new class of functions to the task of modeling the price of call options. Experiments show improvements on regressing the price of call options using the new types of function classes that incorporate the a priori con(cid:173) straints.