Loglinear modeling for three-dimensional contingency tables was used with data from 14 rainfall stations located in Alentejo and Algarve region, southern of Portugal, for short term prediction of drought severity classes. Loglinear models were fitted to drought class transitions derived from Standardized Precipitation Index (SPI) time series computed in a 12-month time scale. Quasi-association loglinear models proved to be the most adequate in fitting all the 14 data series. Odds and respective confidence intervals were calculated in order to understand the drought evolution and to estimate the drought class transition probabilities. The validation of the predictions was performed for the 2004-2006 drought, particularly for periods when the drought was initiating and establishing, and when it was dissipating. Despite the contingency tables of drought class transitions present a strong diagonal tendency, results of three-dimensional loglinear modeling present good results when comparing predicted and observed drought classes with 1 and 2 months lead for those 14 sites. Only for a few cases predictions did not fully match the observed drought severity, mainly for 2-month lead and when the SPI values are near the limit of the severity class. It could be concluded that loglinear prediction of drought class transitions is a useful tool for short term drought warning. ª
In this paper the distribution of the sum of independent 1 variables with different scale parameters is obtained by direct integration, without involving a series expansion. One of its particular cases is the distribution of the product of some particular independent Beta variables. Both distributions are obtained in a concise manageable form readily applicable to research work in the area of multivariate statistics distributions. The exact distribution of the generalized Wilks' 4 statistic is then obtained as a direct application of the results.1998 Academic Press AMS 1991 subject classification numbers: 62E15, 62H10.
In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks L statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution. r
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