Prediction of water shortage loss in situations with small samples based on an improved Gumbel copula

Prediction of water shortage losses is of great importance for water resources management. A new mathematical expression of water shortage loss was proposed in order to describe the random uncertainty and economic attributes of water resources. Then, Gumbel copula with a new method of parameter estimation was introduced to model the joint probabilistic characteristics for water supply and water use in situations when sufficient data is unavailable. The new parameter estimation method requires only the minimum and maximum values of two variables. The improved Gumbel copula was proved to be reliable based on the RMSEs (root mean square error) and AICs (Akaike information criterion), statistical tests and upper tail dependence tests. The potential water shortage losses for all the districts of Tianjin were predicated. The water shortage loss in the Urban district is highest (7.02 billion CNY), followed by the new district of Binhai and Wuqing district, while those in the Baodi district and Ji County are very small.