We consider the response of a uniformly accelerated monopole detector that is coupled non-linearly to the nth power of a quantum scalar field in (D + 1)-dimensional flat spacetime. We show that, when (D + 1) is even, the response of the detector in the Minkowski vacuum is characterized by a Bose-Einstein factor for all n. Whereas, when (D + 1) is odd, we find that a Fermi-Dirac factor appears in the detector response when n is odd, but a Bose-Einstein factor arises when n is even. We emphasize the point that, since, along the accelerated trajectory, the Wightman function and, as a result, the (2n)-point function satisfy the Kubo-Martin-Schwinger condition (as required for a scalar field) in all dimensions, the appearance of a Fermi-Dirac factor (instead of the expected Bose-Einstein distribution) for odd (D + 1) and n reflects a peculiar feature of the detector rather than imply a fundamental change in field theory.