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For an unknown continuous distribution on a real line, we consider theapproximate estimation by the discretization. There are two methods for thediscretization. First method is to divide the real line into several intervalsbefore taking samples ("fixed interval method") . Second method is dividing thereal line using the estimated percentiles after taking samples ("movinginterval method"). In either way, we settle down to the estimation problem of amultinomial distribution. We use (symmetrized) $f$-divergence in order tomeasure the discrepancy of the true distribution and the estimated one. Ourmain result is the asymptotic expansion of the risk (i.e. expected divergence)up to the second-order term in the sample size. We prove theoretically that themoving interval method is asymptotically superior to the fixed interval method.We also observe how the presupposed intervals (fixed interval method) orpercentiles (moving interval method) affect the asymptotic risk.

Estimation of a Continuous Distribution on a Real Line by Discretization Methods -- Complete Version--