**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**

By

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.