, Volume 30, Issue 1, pp 21–29 | Cite as

Consistency of a nonparametric estimation of a density functional

  • I. A. Ahmad
  • P. -E. Lin


LetX be a random variable with distribution functionF and density functionf. Let ϕ and ψ be known measurable functions defined on the real lineR and the closed interval [0, 1], respectively. This paper proposes a smooth nonparametric estimate of the density functional\(\theta = \int\limits_R \phi (x) \psi \left[ {F (x)} \right]f^2 (x) dx\) based on a random sampleX1, ...,X n fromF using a kernel functionk. The proposed estimate is given by\(\hat \theta = (n^2 a_n )^{ - 1} \mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n \phi (X_i ) \psi \left[ {\hat F (X_i )} \right]k\left[ {(X_i - X_j )/a_n } \right]\), where\(\hat F(x) = n^{ - 1} \mathop \sum \limits_{i = 1}^n K\left[ {(x - X_i )/a_n } \right]\) with\(K (w) = \int\limits_{ - \infty }^w {k (u) } du\). The estimate\(\hat \theta \) is shown to be consistent both in the weak and strong sense and is used to estimate the asymptotic relative efficiency of various nonparametric tests, with particular reference to those using the Chernoff-Savage statistic.


Asymptotic Normality Nonparametric Estimation Dominant Term Lebesgue Dominate Convergence Theorem Weak Consistency 
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Copyright information

© Physica-Verlag 1983

Authors and Affiliations

  • I. A. Ahmad
    • 1
  • P. -E. Lin
    • 2
  1. 1.Department of Mathematical SciencesMemphis State UniversityMemphisUSA
  2. 2.Department of StatisticsFlorida State UniversityTallahasseeUSA

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