Stochastic ordering of extreme order statistics in Archimax copula

Abstract

An extension of Archimax copula class in more than two random variables ( Multivariate ) was introduced in (Jágr 2011) for describing dependency structures among random variables in higher dimension, and some properties of Archimax copula were explored in (Charpentier et al. 2014). In this article, some results for stochastic ordering of extreme order statistics in (Li and Fang 2015) are generalized and proved in Archimax copula. Stochastic ordering of sample extremes for PHR models is generalized and proved in Archimax copula. Examples with graphical illustrations are also presented.

Figures (7)

• Figure 1: Graphs of (A) $\frac{t \phi^{\prime}(t)}{\phi(t)} = -\frac{t^{\frac{1}{\theta}}}{\theta}, \, \theta=4, \,A=4^{\frac{1}{4}-1}$ (B) $\frac{F_{X_{4:4}}}{F_{X_{3:4}}}$ (C) $\frac{F_{X_{5:5}}}{F_{X_{4:4}}}.$
• Figure 2: Graphs of (A) $\frac{t \phi^{\prime}(t)}{1- \phi(t)} = -\frac{t^{\frac{1}{\theta}}}{\theta (e^{t^{\frac{1}{\theta}}}-1)},\, \theta=8,\,A=4^{\frac{1}{8}-1}$ (B) $\frac{1-F_{X_{4:4}}}{1-F_{X_{3:4}}}$ (C) $\frac{1-F_{X_{5:5}}}{1-F_{X_{4:4}}}.$
• Figure 3: Graphs of (A) $\frac{t \phi^{\prime\prime}(t)}{ \phi^{\prime}(t)} =\frac{1-t^{\frac{1}{\theta}}}{\theta}-1,\, \theta=5,\, A=4^{\frac{1}{5}-1}\,\,\,\,\,$(B) $\frac{f_{X_{4:4}}}{f_{X_{3:4}}}$ (C) $\frac{f_{X_{5:5}}}{f_{X_{4:4}}}.$
• Figure 4: Graphs in (A) $\frac{t\phi^{\prime}(t)}{\phi(t)} = \frac{\theta}{1+t}-\theta$, $\theta=4,\, A=4^{\frac{1}{4}-1}$ (B) $\frac{1-F_{X_{1:4}}}{1-F_{X_{1:5}}}$ (C) $\frac{1-F_{X_{2:4}}}{1-F_{X_{1:4}}}.$
• Figure 5: Graphs in (A) $\frac{t \phi^{\prime}(t)}{(1- \phi(t))} = \frac{t\theta}{1+t-(1+t)^{\theta+1}},\, \theta=8,\, A=4^{\frac{1}{8}-1}$ (B) $\frac{F_{X_{1:4}}}{F_{X_{1:5}}}$ (C) $\frac{F_{X_{2:4}}}{F_{X_{1:4}}}.$