Some results on maximum of expected sums of sequential minima

Let X1,…,Xn be independent and identically distributed continuous nonnegative random variables and mn be the sample minimum. In the present paper, we propose an approach, which maximizes the value E(∑i=knmin{Xi,…,Xn}) and makes it greater than E(∑i=1nmi|X1=x1,…,Xk−1=xk−1). We further apply this approach to the standard uniform and exponential distributions and for finding the corresponding values x1,…,xk−1 solve proper algebraic equations by numerical methods.