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Second Order Efficient Estimating a Smooth Distribution Function and its Applications

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  • Sam Efromovich

    (The University of New Mexico)

Abstract

Consider a problem of estimation of a cumulative distribution function of a random variable supported on a finite interval, with a circular random variable being a particular case. It is well known that empirical (sample) distribution is asymptotically first order efficient, that is, its mean squared error converges with optimal rate and constant. However, the estimator is discontinuous. Thus, is it possible to suggest a better estimator for the case of a smooth distribution, in particular an analytic one? The answer is “yes” and, interestingly, the derivative of the estimator suggested is an efficient estimate of an underlying probability density. Moreover, Monte Carlo simulations reveal that an adaptive estimator mimicking the second order efficient estimator have attractive properties for practically important small samples.

Suggested Citation

  • Sam Efromovich, 2001. "Second Order Efficient Estimating a Smooth Distribution Function and its Applications," Methodology and Computing in Applied Probability, Springer, vol. 3(2), pages 179-198, June.
    • Handle: RePEc:spr:metcap:v:3:y:2001:i:2:d:10.1023_a:1012257227215
    DOI: 10.1023/A:1012257227215
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    Cited by:

    1. Stefano Maria Iacus & Davide La Torre, 2002. "On fractal distribution function estimation and applications," Departmental Working Papers 2002-07, Department of Economics, Management and Quantitative Methods at Università degli Studi di Milano.

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