On maximum variance of kth records

We consider i.i.d. samples of size n with continuous parent distributions and finite variances. Klimczak and Rychlik (2004) provided sharp bounds for the variances of kth record values in population variance units. The bounds are expressed in terms of the maximum of an analytic function over interval (0, 1). We prove that there exists a point x = x(k, n) such that the function strictly increases in (0, x) and strictly decreases in (x, 1). Consequently the maximum point is unique and it is the zero of the function derivative. We solve this problem for 2≤k≤max{2,n2+4n3n+4}$2\le k\le \max \lbrace 2,\frac{n^{2}+4n}{3n+4}\rbrace$ and n ⩾ 1 applying the variation diminishing property of the power functions.