Let X 1 , , X n be independent exponential random variables with X i having hazard rate λ i , i = 1, , n. Let λ = (λ 1 , , λ n ). Let Y 1 , , Y n be a random sample of size n from an exponential distribution with common hazard rate λ = n i = 1 λ i /n. The purpose of this paper is to study stochastic comparisons between the largest order statistics X n : n and Y n : n from these two samples. It is proved that the hazard rate of X n : n is smaller than that of Y n : n . This gives a convenient upper bound on the hazard rate of X n : n in terms of that of Y n : n . It is also proved that Y n : n is smaller than X n : n according to dispersive ordering. While it is known that the survival function of X n : n is Schur convex in λ, Boland, El-Neweihi and Proschan [J. Appl. Probab. 31 (1994) 180-192] have shown that for n > 2, the hazard rate of X n : n is not Schur concave. It is shown here that, however, the reversed hazard rate of X n : n is Schur convex in λ.