Is it possible to test if a power of the density is integrable?

Abstract

It is not clear what $L_p$-distance should be used to measure the error in density estimation. Whereas the $L_1$-distance makes sense for any pair of densities, the $L_2$-distance is always more tractable mathematically and the $L_\infty$-distance provides a uniform metric; however, for the latter two distances we should check first if the unknown density is square integrable or (essentially) bounded, respectively. We prove that there is no test for such null hypotheses making the right decision with arbitrarily high probability and based on a finite number of data.