Tobit Model
The Tobit model is a statistical model proposed by James Tobin (1958) to describe the relationship between a non-negative dependent variable y i {\displaystyle y_{i}} y_{i} and an independent variable (or vector) x i {\displaystyle x_{i}} x_{i}.[1] The term Tobit was derived from Tobin’s name by truncating and adding -it by analogy with the probit model.[2] The Tobit model is distinct from the truncated regression model, which is in general different and requires a different estimator.[3] The model supposes that there is a latent (i.e. unobservable) variable y i * {\displaystyle y_{i}^{*}} y_i^*. This variable linearly depends on x i {\displaystyle x_{i}} x_{i} via a parameter (vector) ß {\displaystyle \beta } \beta which determines the relationship between the independent variable (or vector) x i {\displaystyle x_{i}} x_{i} and the latent variable y i * {\displaystyle y_{i}^{*}} y_i^* (just as in a linear model). In addition, there is a normally distributed error term u i {\displaystyle u_{i}} u_{i} to capture random influences on this relationship. The observable variable y i {\displaystyle y_{i}} y_{i} is defined as the ramp function: equal to the latent variable whenever the latent variable is above zero, and zero otherwise. …

Understanding dependence structure among extreme values plays an important role in risk assessment in environmental studies. In this work we propose the $\chi$ network and the annual extremal network for exploring the extremal dependence structure of environmental processes. A $\chi$ network is constructed by connecting pairs whose estimated upper tail dependence coefficient, $\hat \chi$, exceeds a prescribed threshold. We develop an initial $\chi$ network estimator and we use a spatial block bootstrap to assess both the bias and variance of our estimator. We then develop a method to correct the bias of the initial estimator by incorporating the spatial structure in $\chi$. In addition to the $\chi$ network, which assesses spatial extremal dependence over an extended period of time, we further introduce an annual extremal network to explore the year-to-year temporal variation of extremal connections. We illustrate the $\chi$ and the annual extremal networks by analyzing the hurricane season maximum precipitation at the US Gulf Coast and surrounding area. Analysis suggests there exists long distance extremal dependence for precipitation extremes in the study region and the strength of the extremal dependence may depend on some regional scale meteorological conditions, for example, sea surface temperature. …