By Synowka-Bejenka E., Zontek S.

Within the paper the matter of simultaneous linear estimation of mounted and random results within the combined linear version is taken into account. an important and adequate stipulations for a linear estimator of a linear functionality of mounted and random results in balanced nested and crossed type versions to be admissible are given.

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**Extra info for A characterization of admissible linear estimators of fixed and random effects in linear models**

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Let F be any collection of operators on an n-dimensional space over a field whose characteristic is either zero or larger than n/2. Then F is triangularizable if and only if trace is permutable on :F. Proof. 4, the hypothesis extends to the algebra A generated by:F. 1. We must show that the diagonal blocks Aii all act on one-dimensional spaces M. Assume otherwise. 1, say J], corresponds to equal diagonal blocks {Aii : i E JJ} acting on spaces of dimension at least 2. Thus the number m of elements in J 1 is no greater than n/2.

As shown in the case of Bin the preceding paragraph, it follows that K is not invariant under BI and there is an A2 in No such that A2BIX2 i= 0 for some X2 E K. Choose P2 such that A2Bl +p2(A 2, BdA2 is in N, and let B2 = A2BI +p2(A2, BdA2. Then A3B2X3 i= 0 for some A3 E No and X3 E K. Continue in this manner to obtain and vectors {XI,X2, ... ,Xn+l}. Then An+IAn ... A2AIBxn+1 = An+lBnxn+1 i= O. However, No is a triangularizable collection of nilpotent transformations on a space of dimension (n + 1), so the product of (n + 1) members of No is O.

4 Notes and Remarks 41 Proof. 8). 4)). We consider two different cases. First, if P and Q operate on a space of dimension 2, we can choose a similarity such that P = (~ ~ ). PQ - QP = With respect to the same basis, let Q (~c ~), so (PQ - QP)2 = = (~ ~ ). Then 0 implies that at least one of b and c is O. Thus Q is upper or lower triangular, and {P, Q} is triangularizable. If P and Q act on a space of dimension greater than 2, note that the operator R = PQP + (I - P)(I - Q)(I - P) commutes with both P and Q (as seen by an easy direct computation).

### A characterization of admissible linear estimators of fixed and random effects in linear models by Synowka-Bejenka E., Zontek S.

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