Orderings of extremes from dependent Gaussian variables with Archimedean copula under simple tree order restrictions

Abstract

Suppose the random variables $X_1,\dots ,X_{n+1}$ follow normal distributions $N({\mu }_i,{\sigma }_i^2)$, $i=1,\dots ,n+1$, and their dependence is modelled by Archimedean copula with generator ϕ. Then, the following results are established for the largest order statistics: (1) Suppose $\frac{t{\varphi }^{\prime }(t)}{1-\varphi (t)}$ and $\frac{{\mathrm{\Phi }}^{-1}(\varphi (t)){\mathrm{\Phi }}^{\prime }({\mathrm{\Phi }}^{-1}(\varphi (t)))}{t{\varphi }^{\prime }(t)}$ are increasing in t>0, then, if ${\mu }_1=\cdots ={\mu }_{n+1}$ and ${\sigma }_{n+1}\ge {\sigma }_i$ for $i=1,\dots ,n,$ we have $X_{n:n}{\le }_{hr}{X}_{n+1:n+1}$; (2) Suppose $\frac{t{\varphi }^{\prime }(t)}{\varphi (t)}$ is decreasing and $\frac{{\mathrm{\Phi }}^{-1}(\varphi (t)){\mathrm{\Phi }}^{\prime }({\mathrm{\Phi }}^{-1}(\varphi (t)))}{t{\varphi }^{\prime }(t)}$ is increasing in t>0, then, if ${\mu }_1=\cdots ={\mu }_{n+1}$ and ${\sigma }_{n+1}\le {\sigma }_i$ for $i=1,\dots ,n,$ we have $X_{n:n}{\le }_{rh}{X}_{n+1:n+1}$; (3) Suppose $\frac{t{\varphi }^{\prime }(t)}{1-\varphi (t)}$ and $\frac{{\mathrm{\Phi }}^{\prime }({\mathrm{\Phi }}^{-1}(\varphi (t)))}{t{\varphi }^{\prime }(t)}$ are increasing in t>0, then, if ${\sigma }_1=\cdots ={\sigma }_{n+1}$ and ${\mu }_{n+1}\ge {\mu }_i$ for $i=1,\dots ,n,$ we have $X_{n:n}{\le }_{hr}{X}_{n+1:n+1}$; (4) Suppose $\frac{t{\varphi }^{\prime }(t)}{\varphi (t)}$ is decreasing and $\frac{{\mathrm{\Phi }}^{\prime }({\mathrm{\Phi }}^{-1}(\varphi (t)))}{t{\varphi }^{\prime }(t)}$ is increasing in t>0, then, if ${\sigma }_1=\cdots ={\sigma }_{n+1}$ and ${\mu }_{n+1}\le {\mu }_i$ for $i=1,\dots ,n,$ we have $X_{n:n}{\le }_{rh}{X}_{n+1:n+1}$. Analogous results are then established for smallest order statistics as well. In addition, we present some numerical examples to illustrate all the results established here. Finally, some concluding remarks are made.

Keywords:

• Archimedean copula
• hazard rate order
• reversed hazard rate order
• normal distribution
• simple tree order restriction

2000 MSC:

• 60E15
• 62G30
• 62D05

Acknowledgments

The last author thanks the National Sciences and Engineering Research Council of Canada for supporting this research.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the Anhui Provincial Philosophy and Social Science Planning Project (No. AHSKF2021D30).