356 C. R. HENDERSON 



Consequently let us consider some other criterion of a "best" index. We shall 

 use as our criterion of "best" that index from the class of linear functions of 

 the sample which is unbiased (coefBcients of all b^s, = in Ed) and for which 

 E{d— Oy is a minimum. E denotes expected value. Consequently E{d — dY 

 denotes the average in repeated sampling of the squared deviations of the 

 index of 6 about the true value of d. When the i's are unknown, the same 

 criterion of best is applied to them, that is, minimum E{b — bY for unbiased 

 estimates {Eb = b) which are linear functions of the sample. It turns out 

 that minimization of E{d — Oy and maximization of ree lead to identical in- 

 dexes. Hence the assumption of normality is not essential to construction 

 of selection indexes as now used. 



It must be obvious that the selection index method just described is very 

 laborious when a number of different 6 need to be estimated, for the solution 

 to a set of simultaneous equations is required for each 6. In practice this diffi- 

 culty is avoided to a certain extent by choosing arbitrarily only a few sources 

 of information to be employed in selection. This is not a wholly satisfactory 

 solution, for in most cases if the number of different indexes is not to be en- 

 tirely too large, information must be rejected which could add at least a 

 little to the accuracy of the index. 



By means of a simple modification it becomes necessary to solve only one 

 set of equations no matter how many B are estimated from a particular set 

 of data, and precisely the same index as in the conventional method is ob- 

 tained. Using the same notation as before, the index for 6 is now 



where the Cs are the solution to a set of equations identical to set (1) except 

 that the right members are W\, . . . , Wn rather than a^y^, . . . , agy^.. Con- 

 sequently once the C's are computed, any number of 0's can be estimated 

 simply by taking the appropriate linear function of the C's. 



More tedious computations result if the b's are not known. One solution 

 is of the following general form. In order that each d be unbiased it is neces- 

 sary that the K's have these restrictions imposed: 



A'l.Vn + A'2.V]2 + • • • + K^Xix = 



A'i.r2i + A'2.Y22 + . . . + A.vA'l.V = 



(2) 



A'l-Vp, -f A'2-Vp2 + • • • + A.Y.TpTv = 



Subject to these restrictions the values of the A's which minimize E{d — dY 

 can then be found. 



If we wish to obtain estimates of the i's which are unbiased and have 

 minimum E(J) — bY, we impose the restrictions of equations (2) except that 



