2. Unbiasedness in point estimation . Suppose that 'I is a continuous real-valued function defined over n which is not constant in any open subset of n, and that the expectation h(O) = Eg8(X) is a continuous function of 0 for every estimate 8( X) of '1(0). Then (11) is a necessary and sufficient condition for 8( X) to be unbiased when the loss function is the square of the error. [Unbiasedness implies that '1 2(0') - '1 2(0) 2h(0)[y(0') - '1(0)] for all 0,0'. If 0 is neither a relative minimum or maximum of 'I, it follows that there exist points 0' arbitrarily close to 0 both such that '1(0) + y( 0') and 2h(0), and hence that '1(0) = h(O). That this equality also holds for an extremum of 'I follows by continuity, since 'I is not constant in any open set.]