**Sankhya:
The Indian Journal of Statistics**

2005, Volume 67, Pt. 4, 785--794

**Optimal Designs for Estimation of
Ratio of Variance Components in Diallel Crosses**

Himadri Ghosh, I.A.S.R.I., New Delhi,
India

Ashish Das, Indian Statistical Institute, New Delhi, India

Chand K. Midha, The University of Akron, Akron, USA

SUMMARY. The results on optimal designs for diallel crosses are presently available for the standard fixed effects linear model. In some cases, however, parental lines may be randomly selected from a population of lines resulting in a random effects model. Ghosh and Das (2003) discussed $A$-optimal designs in this context for estimation of heritability. In this paper we first propose an unbiased estimator of the ratio of the variance components which has a one-to-one relation with heritability. We then obtain an expression of the variance of this unbiased estimator of the ratio of variance components. Through minimization of the variance we obtain optimal designs and show certain connections with the optimization problem under the fixed effects model.

*AMS (2000) subject classification. *62K10, 62K05.

*Key words and phrases. *$A$-optimality, variance components,
asymptotically unbiased estimator, heritability.