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An Esséen-type inequality for probability density functions, with an application

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  • Roussas, George G.

Abstract

In this note, an upper bound is provided for the supremum of the absolute value of the difference of the probability density functions of two k-dimensional random vectors. The bound involves integrals of the absolute value of the characteristic functions of the random vectors, and shares a general similarity with a bound obtained by Sadikova for distribution functions of two-dimensional random vectors. Sadikova's paper provided the impetus for this note. Special cases are considered, and an application is presented, regarding consistency of a kernel estimate in the framework of associated random variables.

Suggested Citation

  • Roussas, George G., 2001. "An Esséen-type inequality for probability density functions, with an application," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 397-408, February.
  • Handle: RePEc:eee:stapro:v:51:y:2001:i:4:p:397-408
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    References listed on IDEAS

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    1. Roussas, G. G., 1994. "Asymptotic Normality of Random Fields of Positively or Negatively Associated Processes," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 152-173, July.
    2. Bagai, Isha & Prakasa Rao, B. L. S., 1991. "Estimation of the survival function for stationary associated processes," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 385-391, November.
    3. Roussas, George G., 1995. "Asymptotic normality of a smooth estimate of a random field distribution function under association," Statistics & Probability Letters, Elsevier, vol. 24(1), pages 77-90, July.
    4. Roussas, George G., 1991. "Kernel estimates under association: strong uniform consistency," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 393-403, November.
    5. Isha Bagai & B. Prakasa Rao, 1995. "Kernel-type density and failure rate estimation for associated sequences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(2), pages 253-266, June.
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    Cited by:

    1. Prakasa Rao, B. L. S., 2002. "Another Esséen-type inequality for multivariate probability density functions," Statistics & Probability Letters, Elsevier, vol. 60(2), pages 191-199, November.
    2. Chang, Wan-Ying & Richards, Donald St.P., 2009. "Finite-sample inference with monotone incomplete multivariate normal data, I," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1883-1899, October.
    3. Li, Yongming & Yang, Shanchao & Wei, Chengdong, 2011. "Some inequalities for strong mixing random variables with applications to density estimation," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 250-258, February.

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