# Analyzing Invariance Properties - Using Reflector Matrices

Browne and Shapiro (1989) give several corollaries which establish properties of reflector matrices implied by various types of scale invariances. Their results include (Shapiro & Browne, 1989, page 8).

Corollary 1.1 If S(q) is invariant under G1*, then the sum of the diagonal elements of is zero.

Corollary 1.2 If S(q) is invariant under G2*, then all diagonal elements of are zero.

These corollaries provide convenient devices for falsifying the dual assertion that a minimum has been obtained for a given discrepancy function, and that the fitted model possesses an invariance property.

Specifically, for a given discrepancy function, the reflector matrix is computed after convergence. If the trace of the reflector matrix is not zero, then either a minimum has not been obtained, or the model is not invariant under a constant scaling factor, or both. Then the individual diagonal elements of the reflector matrix are examined. If they are not all zero, one can conclude that either a minimum has not been obtained, or the model is not invariant under changes of scale, or both.

From a practical standpoint, one must remember issues of machine precision. One would only expect the above criteria to be met to an acceptable level of machine precision. Consequently, SEPATH, besides printing the reflector matrix, also reports (1) the trace of the reflector matrix, and (2) the largest absolute value on the diagonal of the reflector matrix.