3), it can be stated that weekly flow series of the Canadian rivers under question obey the
two-parameter Gamma pdf. The underlying dependence structure of weekly flow series was investigated through week-by-week standardization resulting into weekly SHI sequences. The weekly SHI sequences were subjected to autocorrelation analysis to uncover the presence of Markovian or other higher order dependence. The values of ρ1 ( Table 2) in all rivers are large thus suggesting a strong dependence in successive occurrences of flows. To discern the underlying dependence structure, the values of autocorrelations PD0332991 cell line at lag-1 (ρ1) and lag-2 (ρ2) in weekly SHI sequences ( Table 2) were used to estimate the parameters by fitting ARMA class of models ( Box and Jenkins, FRAX597 concentration 1976). The ARMA models tended to fit AR-1 (autoregressive order-1), AR-2, and ARMA (1,1) dependence structures suggesting dependence terms extending up to the second, and even higher orders in some cases ( Table 2). After fitting the potential models as stated above to the weekly SHI sequences, the autocorrelation function of the residuals was also computed. The Portmanteau statistic based on first 25 autocorrelations
of the residuals formed the basis for suggesting the suitable structure of the model ( Table 2, last column). In particular, rivers in northern Ontario showed dependence structure beyond AR-2, which is comprehensible in view of the significant storage effects caused by the presence of a large number of lakes in watersheds of this region. In a nutshell and as a first approximation of dependence in successive weekly flows, it would be prudent to regard such a dependence to influence flows up to 2 weeks and hence the prediction model for drought length on weekly time scale should be capable to embed the second order dependence. The Markov Chain-2 offers such a capability and thus it should be considered suitable for modeling drought lengths on weekly time scale. The extreme number theorem was used for the prediction of E(LT) using SHI sequences of appropriate time scale. Succinctly, the extreme number theorem culminates in PIK3C2G the following equations
for the prediction of E(LT) ( Sen, 1980a) equation(1) P(LT=j)=exp[−T q (1−r) rj−1][exp T q 2(1−r) rj−1−1]P(LT=j)=exp[−T q (1−r) rj−1][exp T q (1−r)2 rj−1−1] equation(2) E(LT)=∑j=1∞j P(LT=j) where j stands for length of the drought duration and takes on values 1, 2, 3,… up to infinity, q stands for the probability of drought at the given truncation level, say z0 and T is the time equivalent to the sample size of the data involved in the drought analysis. The value of r (first order conditional probability) representing dependence characteristics of a drought is related to ρ1 as shown by Sen (1977) through the following relationship equation(3) r=q+12πq∫0ρ1[exp−z02/(1+ν)](1−ν2)−0.5dνwhere v is a dummy variable for integration. The integral in Eq.