Development of regional regression relationships with censored data

When no discharge record is available for a site, a regional regression relationship can be used to estimate low-flow quantiles. Problems arise in the derivation of such models when some at-site quantile estimates are reported as zero. One concern is that quantile estimates reported as zero may be in the range from zero to the measurement threshold. A second concern is that a logarithmic transformation cannot be used with zero quantile estimates, so traditional log linear least squares estimators cannot be computed. This paper uses visual examples and Monte Carlo simulation to compare the performance of techniques for estimating the parameters of a regional regression model when some at-site quantile estimates are zero. Ordinary least squares (OLS) techniques employed in practice include adding a small constant to all at-site quantile estimates (denoted OLSC), or neglecting all observation reported as zero (denoted OLSD). OLSC and OLSD performed poorly compared to the use of a Tobit model, which is a maximum likelihood estimator (MLE) procedure that represents the below threshold estimates as a range from zero to the threshold level. For a small amount of censoring, the OLSD method can be acceptable. A weighted Tobit model that accounts for the heteroscedasticity of the residuals in the regression model provided relatively little gain over the ordinary Tobit model.