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The Rate Distortion Test of Normality

We use techniques from rate distortion theory in testing normality. The idea is first to do optimal compression with respect to squared Euclidean distance and then use information divergence of the compressed empirical distribution from the compressed Gaussian as statistic. We can analyze how the test performs at different rates. At low rate one gets a test that is efficient against other distributions in the Gaussian location family. If the rate is fixed at a positive value or the rate increases slower than the logarithm of the sample size as the sample size tends to infinity then the rate distortion test will asymptotically as as powerful as the likelihood ratio test. The test will first get efficient against elements in the exponential families generated by moment constraints of low order.