Sankhya: The Indian Journal of Statistics

2003, Volume 65, Pt. 2, 317--332

Bounds On The Kolmogorov Distance Of A Mixture From Its Parent Distribution


BAHA-ELDIN KHALEDI, Statistical Research Center, Tehran and Razi University, Kermanshah, Iran and MOSHE SHAKED, University of Arizona, Tucson, USA

SUMMARY: Consider a mixture \(G(\cdot)=\int_SF_\theta(\cdot)\,dH(\theta)\). In this  paper we derive some bounds on the uniform (Kolmogorov) distance \(\Delta(F_{\theta_0},G)\equiv\sup_x|F_{\theta_0}(x)-G(x)|\) for some convenient choices of \(\theta_0\). In particular, we identify an optimal \(\theta_0\). We illustrate the results by some examples, and show that these new bounds can often be computed easily, and that they  improve some known bounds in many instances. Some applications in reliability  theory are also described.

AMS (1991) subject classification. 60E15, 62E17, 60K10.

Key words and phrases. Hazard rate stochastic order, relative aging stochastic order, TP\(_2\) and RR\(_2\) properties, length-biased distribution.

Full paper (PDF)