A Bivariate Mixed Lognormal Distribution with an Analysis of Rainfall Data

The paper proposes a bivariate Mixed lognormal (Δ2) distribution as a probability model for representing rainfalls, containing zeros, measured at two monitoring sites and provides the maximum-likelihood (ML) estimates or ML estimating equations of the parameters for the Δ2 distribution. The distribution extends the univariate mixed lognormal distribution to the bivariate case. Two procedures for model selection are proposed: the first is the use of statistical test and the second is done by minimizing the Akaike's information criterion (AIC). An illustration for an analysis of the AMeDAS (Automated Meteorological Data Acquisition System) daily rainfall dataset observed in the summer half-year (May–October) of 1988 is given. A Δ2 distribution is fitted to the bivariate data obtained from Tokyo and each of the other 1048 monitoring sites. Among the 1048 cases investigated, the agreement between the models selected by the test (5% significance level) and the minimum AIC procedures is 846 cases (80.7%). A more careful study for Tokyo and each of the other 46 prefecture capitals, however, gives an interpretation of the complete agreement between two procedures if the difference of AICs is considered to be insignificant when it is smaller than 2.