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Novel physical insights are provided into the performances of strictly linear (SL) and widely linear (WL) estimators of the generality of complex-valued data, both proper (second order circular) and improper (second-order noncircular). This is achieved by first performing a novel complementary mean square error (CMSE) analysis, in order to quantify the degrees of improperness (second order noncircularity) of the SL and WL estimation errors. The exact bounds on the CMSE difference between the SL and WL estimators are investigated to show that only a joint consideration of the standard MSE analysis and the proposed CMSE analysis has enough degrees of freedom for a rigorous account of the performance of WL and SL estimators. This also makes it possible to rigourously quantify the contributions to the WL performance advantage from the individual real and imaginary channels, an important finding not …