This assumption also yields marginally conservative values of the deficit-volume which is a desirable feature in the design of water resources systems towards ameliorating the drought conditions. It is worthy to mention that Millan and Yevjevich (1971) developed the regression equations for predicting E(LT) and E(MT) which were also tested for the annual and monthly hydrological droughts using Canadian

river flows. These relationships were found reasonably JQ1 reliable although at times they tended to under predict in the range of 3–10%. As a note in the context of analysis of monthly droughts, it is prudent to mention that the values of ρ1 in the SHI sequences were low suggesting a weak dependence structure. Therefore, the first order Markov chain model (Markov chain-1, Eq. (8)) was tried to estimate E(LT). It was noted that the predictions of LT tended to be almost the same as predicted by the extreme number theorem. However, at times the predictions by the extreme

number theorem tended to be marginally higher than the Markov chain-1 model and also be nearer to the observed counterparts. This observation vindicates the applicability of the extreme number theorem on monthly as well as annual basis. In fact as the name reads “theorem of extremes of random numbers of random variables” essentially is meant for random sequences, which is evidenced by the results in the present case (annual flows). It has the capability to perform reasonably well in the presence of weak dependence structure and for this reason Alectinib order it performed satisfactorily even in monthly streamflow series. It was also observed that when the degree of the first order dependence is remarkable (i.e. ρ1 being above 0.5) then the extreme number theorem breaks down and recourse to the Markov chain models,

among others becomes a necessity. The weekly SHI sequences of rivers with negligible lake effects such as those in Atlantic Canada tended to follow AR-1 process, therefore the extreme number theorem based relationships (Eqs. (1), (2), (3), (4) and (5)) were attempted to model E(LT). In general, such a model resulted in consistent under prediction. As noted earlier, the weekly SHI sequences Plasmin of rivers riddled with significant lake storages tend to obey AR-2 process or even higher order dependence processes ( Table 2). For such rivers, the extreme number theorem does not hold because of a lack of accountability for the second order dependence. Therefore, a second order Markov chain model (Eq. (7)) was envisaged in which the parameters were computed using the counting method ( Sen, 1990 and Sharma and Panu, 2010). The best estimates of the first order probabilities were obtained using the non-standardized weekly flow series ( Table 2).