Second Order Efficient Estimating a Smooth Distribution Function and its Applications

Sam Efromovich ()
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Sam Efromovich: The University of New Mexico

Methodology and Computing in Applied Probability, 2001, vol. 3, issue 2, 179-198

Abstract: 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.

Keywords: adaptation; analytic function; circular random variable; exact constants; nonparametric estimation; probability density; small sample (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1012257227215

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