On efficiency of estimation and testing with data quantized to fixed number of cells

In continuous parametrized models with i.i.d. observations we consider finite quantizations. We study asymptotic properties of the estimators minimizing disparity between the observed and expected frequencies in the quantization cells and asymptotic properties of the goodness of fit tests rejecting the hypotheses when the disparity is large. The disparity is measured by an appropriately generalized &phgr;-divergence of probability distributions so that, by the choice of function &phgr;, one can control the properties of estimators and tests. For bounded functions &phgr; these procedures are robust. We show that the inefficiency of the estimators and tests can be measured by the decrease of the Fisher information due to the quantization. We investigate theoretically and numerically the convergence of the Fisher information. The results indicate that, in the common families, the quantizations into 10–20 cells guarantees ‘practical efficiency’ of the quantization-based procedures. These procedures can at the same time be robust and numerically considerably simpler than similar procedures using the unreduced data.