ISSN:
1572-915X
Keywords:
thinned empirical process
;
point process
;
loglikelihood ratio
;
local asymptotic normality
;
central sequence
;
regular estimators
;
asymptotic efficiency
;
fuzzy set density estimator
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We establish local asymptotic normality of thinned empirical point processes, based on n i.i.d. random elements, if the probability $${\alpha }_{n}$$ of thinning satisfies $${\alpha }_n \to _{n \to \infty } 0,n{\alpha }_n \to _{n \to \infty } \infty$$ . It turns out that the central sequence is determined by the limit of the coefficient of variation of the tangent function. The central sequence depends only on the total number $${\tau }\left( n \right)$$ of nonthinned observations if and only if this limit is 1 or −1. In this case under suitable regularity conditions, an asymptotically efficient estimator of the underlying parameter can be based on $${\tau }\left( n \right)$$ only. An application to density estimation leads to a fuzzy set density estimator, which is efficient in a parametric model. In a nonparametric setup, it can also outperform the usual kernel density estimator, depending on the values of the density and its second derivative.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1009981817526
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