Premium
A bivariate infinitely divisible law for modeling the magnitude and duration of monotone periods of log‐returns
Author(s) -
BarretoSouza Wagner,
Silva Rodrigo B.
Publication year - 2019
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/stan.12166
Subject(s) - mathematics , bivariate analysis , monotone polygon , inverse gaussian distribution , independent and identically distributed random variables , gaussian , random variable , magnitude (astronomy) , inverse , statistics , econometrics , distribution (mathematics) , mathematical analysis , physics , geometry , quantum mechanics , astronomy
The focus of this article is modeling the magnitude and duration of monotone periods of log‐returns. For this, we propose a new bivariate law assuming that the probabilistic framework over the magnitude and duration is based on the joint distribution of ( X , N ), where N is geometric distributed and X is the sum of an identically distributed sequence of inverse‐Gaussian random variables independent of N . In this sense, X and N represent the magnitude and duration of the log‐returns, respectively, and the magnitude comes from an infinite mixture of inverse‐Gaussian distributions. This new model is named bivariate inverse‐Gaussian geometric ( B I G G in short) law. We provide statistical properties of the B I G G model and explore stochastic representations. In particular, we show that the B I G G is infinitely divisible, and with this, an induced Lévy process is proposed and studied in some detail. Estimation of the parameters is performed via maximum likelihood, and Fisher's information matrix is obtained. An empirical illustration to the log‐returns of Tyco International stock demonstrates the superior performance of the B I G G law compared to an existing model. We expect that the proposed law can be considered as a powerful tool in the modeling of log‐returns and other episodes analyses such as water resources management, risk assessment, and civil engineering projects.