A penalty method for nonparametric estimation of the logarithmic derivative of a density function |
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Authors: | Dennis D Cox |
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Institution: | (1) University of Wisconsin, Madison, Wisconsin, USA |
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Abstract: | Summary Given a random sample of sizen from a densityf
0 on the real line satisfying certain regularity conditions, we propose a nonparametric estimator forψ
0=−f
0
′
/f0. The estimate is the minimizer of a quadratic functional of the formλJ(ψ)+∫ψ
2−2ψ′]dFn where λ>0 is a smoothing parameter,J(·) is a roughness penalty, andF
n
is the empirical c.d.f. of the sample. A characterization of the estimate (useful for computational purposes) is given which
is related to spline functions. A more complete study of the caseJ(ψ)=∫d
2ψ/dx2]2 is given, since it has the desirable property of giving the maximum likelihood normal estimate in the infinite smoothness
limit (λ→∞). Asymptotics under somewhat restrictive assumptions (periodicity) indicate that the estimator is asymptotically
consistent and achieves the optimal rate of convergence. This type of estimator looks promising because the minimization problem
is simple in comparison with the analogous penalized likelihood estimators.
This research was supported by the Office of Naval Research under Grant Number N00014-82-C-0062. |
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Keywords: | Nonparametric density estimation roughness penalty spline function |
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