1.10 For any set of numbers x1, ··· , xn and a monotone function h(·), show that the value of a that minimizes
n i=1[h(xi)−h(a)]2 is given by a = h−1 
n i=1 h(xi)/n

. Find functions h that will yield the arithmetic, geometric, and harmonic means as minimizers.

[Hint: Recall that the geometric mean of non-negative numbers is # xi

1/n and the harmonic mean is

(1/n)

(1/xi)

!−1

. This problem, and some of its implications, is considered by Casella and Berger (1992).]