Revisiting Sen's inequalities on order statistics

Sen [1970. A note on order statistics for heterogeneous distributions. Ann. Math. Statist. 41, 2137–2139] derived some inequalities for distributions of order statistics from independent and non-identically distributed (INID) random variables. For this purpose, he used Hoeffding's [1956. On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27, 713–721] inequality concerning distributions of convolutions of Bernoulli random variables. In this note, we present an alternate proof for Sen's inequalities which also leads to various generalizations.