Sharp bounds on the expectations of second record values from symmetric populations

Abstract

Combining the greatest convex minorant approximation (Moriguti, S. (1953). A modification of Schwarz's inequality with applications to distributions. Ann. Math. Statist., 24, 107–113.) with the Hölder inequality, we establish sharp bounds on the expectations of the second record statistics from symmetric populations. We also determine the distributions for which the bounds are attained. The optimal bounds are numerically evaluated and compared with other classical rough ones.

Keywords:

• Record statistics
• Bounds for moments
• Symmetric population
• Hölder inequality
• pth central absolute moment
• Moriguti monotone approximation

Acknowledgements

The problem of the paper was discussed and solved during the stimulating research group meeting at the Banach Center, Polish Academy of Sciences, Warsaw in May 2002. The first author would like to thank the University of Jordan for supporting this research work. The second author was supported by the Polish State Committee for Scientific Research (KBN) Grant no. 5 P03A 012 20. The authors also thank the referees for their useful remarks.

Notes

^†E-mail: [email protected]

Additional information

Notes on contributors

Tomasz Rychlik†

†E-mail: [email protected]