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

Suppose the random variables X1,…,Xn+1 follow normal distributions N(μi,σi2), i=1,…,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 tϕ′(t)1−ϕ(t) and Φ−1(ϕ(t))Φ′(Φ−1(ϕ(t)))tϕ′(t) are increasing in t>0, then, if μ1=⋯=μn+1 and σn+1≥σi for i=1,…,n, we have Xn:n≤hrXn+1:n+1; (2) Suppose tϕ′(t)ϕ(t) is decreasing and …