Estimation of a continuous distribution on the real line by discretization methods

For an unknown continuous distribution on the real line, we consider the approximate estimation by discretization. There are two methods for discretization. The first method is to divide the real line into several intervals before taking samples (“fixed interval method”). The second method is to div...

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Bibliographic Details
Published in:Metrika 2019-04, Vol.82 (3), p.339-360
Main Author: Sheena, Yo
Format: Article
Language:English
Subjects:
Approximation
Asymptotic methods
Asymptotic series
Discretization
Divergence
Economic Theory/Quantitative Economics/Mathematical Methods
Intervals
Mathematics and Statistics
Probability Theory and Stochastic Processes
Statistics
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Summary:For an unknown continuous distribution on the real line, we consider the approximate estimation by discretization. There are two methods for discretization. The first method is to divide the real line into several intervals before taking samples (“fixed interval method”). The second method is to divide the real line using the estimated percentiles after taking samples (“moving interval method”). In either method, we arrive at the estimation problem of a multinomial distribution. We use (symmetrized) f -divergence to measure the discrepancy between the true distribution and the estimated distribution. Our main 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 the moving interval method is asymptotically superior to the fixed interval method. We also observe how the presupposed intervals (fixed interval method) or percentiles (moving interval method) affect the asymptotic risk.
ISSN:0026-1335
1435-926X
DOI:10.1007/s00184-018-0683-y