Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables

Let X₁, ..., X_n be independent exponential random variables with respective hazard rates λ₁, ..., λ_n, and let Y₁, ..., Y_n be independent exponential random variables with common hazard rate λ. This paper proves that X_{2 : n}, the second order statistic of X₁, ..., X_n, is larger than Y_{2 : n}, the second order statistic of Y₁, ..., Y_n, in terms of the likelihood ratio order if and only if λ ≥ frac(1, 2 n - 1) (2 Λ₁ + frac(Λ₃ - Λ₁ Λ₂, Λ₁² - Λ₂)) with Λ_k = ∑_{i = 1}ⁿ λ_i^k, k = 1, 2, 3. Also, it is shown that X_{2 : n} is smaller than Y_{2 : n} in terms of the likelihood ratio order if and only if λ ≤ frac(underover(∑, i = 1, n) λ_i - under(max, 1 ≤ i ≤ n) λ_i, n - 1) . These results form nice extensions of those on the hazard rate order in Pa ̌lta ̌nea [E. Pa ̌lta ̌nea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993-1997].