Right spread order of the second-order statistic from heterogeneous exponential random variables

Let X₁,..., X_n be independent exponential random variables with respective hazard rates λ₁,..., λ_n, and Y₁,..., Y_n be i.i.d. exponential random variables with common hazard rate λ. It is proved that X _2:n, the second order statistic from X₁,..., X _n, is larger than Y_2:n, the second order statistic from Y₁,...,Y_n, with respect to the right spread order if and only if λ≥2n-1/n(n-1)(Σ_i=1ⁿ1/∧-n-1/ ∧(1)) with ∧(1) and Σ_i=1ⁿλ _i and ∧_i = ∧(1) - λ_i, and X _2:n is smaller than Y_2:n with respect to the right spread order if and only if λ ≤ Σ_i=1ⁿ - max _1≤i≤nλ_i/n-1 Further, the case with proportional decreasing hazard rate is also studied, and the results obtained here form nice extensions to some corresponding ones known in the literature.