A New Trivariate Model and Generalized Linear Model for Stochastic Episodes' Duration, Magnitude and Maximum

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Zuniga, Francesco

Issue Date

2021

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Dissertation

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Generalized Linear Model , Hurdle model , Mixed Distribution , Statistics and Data Science

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Abstract

In this dissertation we work with a trivariate model for stochastic events $(N, X, Y)$, where $N$ is the duration, $X$ is the magnitude, and $Y$ is the maximum of an event. We first consider the case, where N has a geometric distribution, $X=\sum_{i=1}^{N}X_i$, and $Y = \bigvee_{i=1}^{N}X_i$, where the $X_i$'s are independent and identically distributed (IID) exponential random variables. Such events arise, for example, when a process is observed above or below a threshold. Examples include heat waves, flood, drought, or market growth or decline periods. In this setting, we extend the IID model to one that incorporates covariates. \\ We prove existence and uniqueness of the maximum likelihood estimators for the parameters, and introduce a new method for checking the goodness of fit of the model to the data. Our goodness of fit method is based on distributional fit of appropriately transformed data. We include a data example from finance to illustrate the modeling potential of this new generalized linear model.\\ In the second part, we extend the model for $(N, X, Y)$ to the case where $N$ has a 1-inflated (or deflated) geometric distribution. Data requiring this extension appear in several areas of applications, including actuarial science, finance, and weather and climate. We provide basic properties and estimation of parameters of this new class of multivariate mixed distributions. Our results include marginal and conditional distributions, joint integral transforms, moments, stochastic representations, estimation and testing. An example from finance illustrates the modeling potential of this new model. \\ Finally, we show an important application of our model to weather data, where the process of interest is total daily precipitation. Here, the random vectors \\ $(N, X, Y)$ describe the duration, the magnitude and the maximum of precipitation events, defined as consecutive days when precipitation is greater than a certain threshold. We fit our model to observational and predicted precipitation data from several global climate models for the time period 1950-2100, with particular attention to the storms generated by the atmospheric rivers. We show that the estimated parameters change to reflect the observed topography and type of storms in the region of interest. Our results provide an insight to the duration and magnitude of the future storms, which in turn provide the quantitative information for the water resources and emergency planning.

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