Ingrid Dæhlen

This article concerns hybrid combinations of empirical and parametric likelihood functions. Combining the two allows classical parametric likelihood to be crucially modified via the nonparametric counterpart, making possible model misspecification less problematic. Limit theory for the hybrid likelihood function is sorted out, also outside of the parametric model conditions. We prove a profiling result as well as limiting behaviour of the maximizer of the hybrid likelihood function. Our results allow for the presence of plug-in parameters in the hybrid and empirical likelihood framework. Furthermore, the variance and mean squared error of these estimators are studied, with recipes for their estimation. The latter is used to define a focused information criterion, which can be used to choose how the parametric and empirical part of the hybrid combination should be balanced. This allows for hybrid models to be fitted in a context driven way.

Environmental risk analyses of sewage systems shall be an integrated part of municipalities’ master plan for wastewater and storm water, as well as applications for permission of outlet to the State Governor. Such analyses are often performed in a manual and ad hoc, yet time-consuming way. Such simplified analyses typically include relatively few pipes, and the probability for different risk categories are often provided as crude estimates only. This approach does not at all harvest the full potential of connections and explanatory power that is contained in all the data available from the pipe system and the environment in combination. In this project, we develop a statistical framework that calculates environmental risk for 6200 sewage pipes based on data available for the municipality. The data set comprises three major sources: 1) Data from the professional VA wiring network system Gemini VA. This data set holds information on the geographical position of pipes, their age, material, diameter and slope, along with a history of maintenance and failures. 2) Data obtained from simulations of the sewage system in the hydraulic-hydrological software tool MIKE+. We perform model simulations for different precipitation events and prolonged dry weather periods. From these simulations we obtain for each pipe values of importance to pollution risk, for example: Overload, pressure, maximum and minimum water flow. 3) Data from environmental monitoring of the storm water system and urban streams in dry as well as wet weather. We measure six parameters: E.coli, total phosphorus, ortophosphate or total reactive phosphorus, total nitrogen, nitrate and ammonia in 400 manholes and stream monitoring locations. These data, called “weather pollution”, are assigned to the closest upstream sewage pipe, which also incorporates system information representative of the total upstream exposure. The analysis produces three main results: 1) Model simulation of a base risk value for each sewage pipe in dry weather or prolonged dry weather periods. The main risk is accumulation of sewage. 2) Calculation of a risk value for each pipe during overflow. The calculations are performed for different weather exposures represented via local precipitation intensity, duration and frequency (IDF) values for selected return periods. 3) Prediction of a “dry weather pollution” base risk value for pipes that have no monitored values, based on feeding their properties to the statistical framework. Most municipalities monitor their streams and lakes through the water areas, and many perform source tracking in the storm water system. If, in addition, a calibrated hydraulic-hydrological model of the sewage system is available, the suggested methodology can provide a more comprehensive analysis which is also more efficient.