A note on why doesn't the choice of performance measure matter?

Choosing an appropriate performance measure is important for fund investors, nevertheless, many researchers find empirically that the choice of measures does not matter because those measures generate identical rank ordering, even though the distribution of fund returns is non-normal. In this paper we certify their findings by proving the monotonicity of several widely used performance measures when the distribution is a location-scale family. The mutual fund monthly return data from 1997 to 2015, together with simulation results, collaborate with our proof.