358 C. R. HENDERSON 



An alternative computational procedure which is less laborious when the O's 

 are few in number, and in particular when the 5's are uncorrelated, involves 

 joint estimation of the i's and 9's by solution of equations (5) to which are 

 added equations (6). 



(6) 



b,Sx,z,-\-. . .+ b,SXj,z,-\-di{Sz,z,+ a'')-\-. . .-^d,(Szl+a"')=Sz,y 

 where 



,-'MI = IIV,l 



an 



d 



SxiZi = > — -^, etc 



These equations are simply least squares equations (the d's are regarded 

 as fixed rather than having a distribution) modified by adding a"-' to certain 

 coefficients. 



SELECTION BY MAXIMUM LIKELIHOOD ESTIMATES 



Now let us assume that the ^'s have the multivariate normal distribution 

 and that the errors are normally and independently distributed. What are 

 the maximum likelihood estimates of the d's and 6's? It just so happens that 

 the estimates which are unbiased and which have minimum E(d — 6)- and 

 E(b — b)- for the class of linear functions of the sample are also the maxi- 

 mum likelihood estimates. Consequently the estimation procedure we have 

 described can be seen to have the following desirable properties: unbiased- 

 ness, maximum relative efficiency of all linear functions of the sample, maxi- 

 mization of genetic progress through selection by truncation when the dis- 

 tributions are normal, properties of maximum likelihood estimates when the 

 distributions are normal, and equations of estimation which can be set up in 

 a routine manner. 



Unknown Variances and Covariances 



An important problem in selection remains unsolved and perhaps there 

 is no practical solution to it. What should be done if the variances and co- 

 variances are unknown? If our sample is so large that estimates of the vari- 

 ances and covariances can be obtained from it with negligible errors, we can 

 use these estimates as the true values. Similarly we may be able to utilize 

 estimates obtained in previous experiments. But if there are no data available 

 other than a small sample, the only reasonable advice would seem to be to 

 estimate the variances from the sample, perhaps modifying these estimates 



