BIAS-CORRECTED MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS OF THE WEIGHTED LINDLEY DISTRIBUTION

This report discusses the calculation of analytic second-order bias techniques for the maximum likelihood estimates (for short, MLEs) of the unknown parameters of the distribution in quality and reliability analysis. It is well-known that the MLEs are widely used to estimate the unknown parameters of the probability distributions due to their various desirable properties; for example, the MLEs are asymptotically unbiased, consistent, and asymptotically normal. However, many of these properties depend on an extremely large sample sizes. Those properties, such as unbiasedness, may not be valid for small or even moderate sample sizes, which are more practical in real data applications. Therefore, some bias-corrected techniques for the MLEs are desired in practice, especially when the sample size is small. Two commonly used popular techniques to reduce the bias of the MLEs, are ‘preventive’ and ‘corrective’ approaches. They both can reduce the bias of the MLEs to order O(n−2), whereas the ‘preventive’ approach does not have an explicit closed form expression. Consequently, we mainly focus on the ‘corrective’ approach in this report. To illustrate the importance of the bias-correction in practice, we apply the bias-corrected method to two popular lifetime distributions: the inverse Lindley distribution and the weighted Lindley distribution. Numerical studies based on the two distributions show that the considered bias-corrected technique is highly recommended over other commonly used estimators without bias-correction. Therefore, special attention should be paid when we estimate the unknown parameters of the probability distributions under the scenario in which the sample size is small or moderate.

Assessment of Flood Risk Under Future Climate Conditions

Global climate change is predicted to have impacts on the frequency and severity of flood events. In this study, output from Global Circulation Models (GCMs) for a range of possible future climate scenarios was used to force hydrologic models for four case study watersheds built using the Soil and Water Assessment Tool (SWAT). GCM output was applied with either the "delta change" method or a bias correction. Potential changes in flood risk are assessed based on modeling results and possible relationships to watershed characteristics. Differences in model outputs when using the two different methods of adjusting GCM output are also compared. Preliminary results indicate that watersheds exhibiting higher proportions of runoff in streamflow are more vulnerable to changes in flood risk. The delta change method appears to be more useful when simulating extreme events as it better preserves daily climate variability as opposed to using bias corrected GCM output.