Extreme Behaviors in Risk Measurement via Joint Tail-Based Cumulative Residual Entropy and Its Generalized Version

Tail risk analysis plays a pivotal strategic role in risk management, particularly in light of economic crises. In this context, the purpose of this paper is to examine the asymptotic properties of Joint Tail-based Cumulative Residual Entropy ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>J</mml:mi> <mml:mi>T</mml:mi> <mml:mi>C</mml:mi> <mml:mi>R</mml:mi> <mml:mi>E</mml:mi> </mml:math> ) in a bivariate setup involving two variables, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>X</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Y</mml:mi> </mml:math> . In this setup, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>X</mml:mi> </mml:math> is considered the variable of interest, while <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Y</mml:mi> </mml:math> serves as the benchmark variable. We provide a generalization of the Joint Tail-based Cumulative Residual Entropy to create a more flexible version that allows for a more comprehensive analysis of extreme risk. This generalization leads to a deeper understanding of the tail relationship between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>X</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Y</mml:mi> </mml:math> and their respective impacts on a specific system. To illustrate our results, we conducted the study under both tail-dependent and tail-independent scenarios. We supplemented our research with practical examples and applied our findings to real-world financial data, employing our proposed non-parametric estimator of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>J</mml:mi> <mml:mi>T</mml:mi> <mml:mi>C</mml:mi> <mml:mi>R</mml:mi> <mml:mi>E</mml:mi> </mml:math> as the basis for our analysis.