Abstract
We study the joint distribution of stochastic events described by (X,Y,N), where N has a 1-inflated (or deflated) geometric distribution and X, Y are the sum and the maximum of N exponential random variables. Models with similar structure have been used in several areas of applications, including actuarial science, finance, and weather and climate, where such events naturally arise. We provide basic properties of this class of multivariate distributions of mixed type, and discuss their applications. Our results include marginal and conditional distributions, joint integral transforms, moments and related parameters, stochastic representations, estimation and testing. An example from finance illustrates the modeling potential of this new model.
Highlights
1 Introduction This paper introduces a new model for stochastic events such as growth/decline periods of a financial return, flood, drought, or a heat wave, among others
I=1 i=1 where the {Xi}i≥1 are independent and identically distributed (IID) exponential random variables given by the probability density function (PDF)
In climate/hydrology, a flood may be described as stage of a stream exceeding a levy, heat wave may be described as consecutive days when the maximum daily temperature exceeds a high threshold, deluge can be thought of as consecutive observations of precipitation exceeding a high threshold
Summary
This paper introduces a new model for stochastic events such as growth/decline periods of a financial return, flood, drought, or a heat wave, among others. Of dealing with extensive zeros in the literature are zero -inflated (ZI) (or zero-adjusted, zero-altered) and hurdle (H) models (see, e.g., Cameron and Trivedi 1998, 2005; Lambert 1992; Mullahy 1986; 1997; Panicha 2018; Zuur et al 2009; Alshkaki 2016; and references therein) These two approaches to account for large number of zeros involve mixture distributions with two components, but they differ in the way that zeros can occur. 2.1 The zero-inflated model The ZI model is a mixture of point mass at zero and a counting random variable N The latter is often chosen to follow a standard discrete distribution such as Poisson, geometric or negative binomial (see, e.g., Mullahy 1986; Lambert 1992). Remark 1 The corresponding mixture representation, connecting the relevant random variables, is as follows: NZI =d JN, where J is a Bernoulli random variable with parameter 1 − q, independent of N
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