Maximum variance of order statistics from symmetric populations revisited

Abstract

We consider i.i.d. samples of size n with symmetric non-degenerate parent distributions and finite variances. Papadatos [A note on maximum variance of order statistics from symmetric populations, Ann. Inst. Statist. Math. 48 (1997), pp. 117–121] proved that the maximal variance of each non-extreme order statistic, expressed in the population variance units, is attained in a one-parametric family of symmetric two- and three-point distributions. The parameters of the extreme variance distributions coincide with the arguments maximizing some polynomials of degree 2n−1 over a finite interval. The bounds for variances are equal to the maximal values of the polynomials. We present a more precise solution to the problem by applying the variation diminishing property of Bernstein polynomials.

Keywords:

• i.i.d. sample
• symmetric distribution
• order statistic
• variance
• Bernstein polynomial
• variation diminishing property

2000 Mathematics Subject Classifications :

• 60E15
• 62G30

Acknowledgements

The authors thank an anonymous referee for careful reading and suggesting numerous corrections. The second author was supported by the Polish Ministry of Science and Higher Education Grant no. N N201 416739.