The balanced one-way random model can be written as; y(,ij) = (mu) + a(,i) + e(,ij) (i = 1,...,I; j = 1,....J),;where the random effects a(,1),...,a(,I) are identically distributed as;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);the random errors e(,11),e(,12),...,e(,IJ) are identically distributed as;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);and a(,1),...,a(,I),e(,11),e(,12),...,e(,IJ) are statistically independent. Let (alpha)(,i) be the realized, but unobservable, value of a(,i). Solutions to the problem of estimating the (alpha)(,i)'s when;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);is known are well-established in the literature. The best linear unbiased estimator of (alpha)(,i) when (rho) is known is:; (alpha)(,i) = (1-(rho))(y(,i.)-y(,0.));We consider the estimation of linear combinations of (mu),(alpha)(,1),...,(alpha)(,I) under the more realistic assumption that (rho) is unknown. The parameter (rho) in(' )(alpha)(,i) is replaced by (rho), where (rho) is an estimator of (rho). Eighteen different estimators of (rho) are considered and are classified into five categories. The corresponding estimators of the (alpha)(,i)'s are evaluated in terms of bias, total bias, total bias, conditional bias, total condition bias, mean squared error, total mean squared error, conditional mean squared error, and total conditional mean squared error. It is found that the problem of individual estimation of the (alpha)(,i)'s is closely related to the problem of simultaneous conditional estimation of the group means (mu)(,i) = (mu)+(alpha)(,i) (i = 1,...,I). Relationships to the James-Stein estimation of the mean vector of a multivariate normal distribution are discussed.