A reference prior is suggested in some multiparameter non-regular cases. It is based on an asymptotic expansion of Lindley's measure of information [\textit{D. V. Lindley}, Ann. Math. Statistics 27, 986-1005 (1956; Zbl 0073.14103)] which is the expected Kullback-Leibler divergence between the posterior and prior. It is assumed that the family of densities depends on two parameters and that there are discontinuities at some points which depend on one of the parameters.