On maximum likelihood estimators for the multisite lag‐one streamflow model: Complete and incomplete data cases

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Abstract
Two practical problems may arise when using the traditional estimators of the parameters in the multisite lag‐one streamflow model: (1) the estimated covariance matrix may not be positive definite thereby preventing its decomposition, a necessary step for synthetic streamflow generation, and (2) if there are missing observations, streamflow data must be truncated to the shortest record resulting in a loss of useful information. This note draws attention to the existence of maximum likelihood estimators which overcome both problems. In the complete data case, explicit maximum likelihood estimators exist. However, in the incomplete data case, iterative maximum likelihood procedures must be used. In particular, the expectation maximization (EM) algorithm is considered in a state‐space framework compatible with the multisite streamflow model; this algorithm has robust convergence properties, is simple to implement, and produces smoothed estimates of the missing data. An example is presented illustrating problems with decomposition of the traditional covariance matrix estimate and also illustrating application of the EM algorithm.

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