Recursive Kernel Density Estimators Under Censoring Verifying an <i>β</i>-mixing Dependence Structure

Mouhamed Mar; Saliou Diouf; El Hadji Deme

doi:doi:10.11648/j.ajtas.20241306.17

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Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure

Mouhamed Mar^*

, Saliou Diouf

, El Hadji Deme

Published in American Journal of Theoretical and Applied Statistics (Volume 13, Issue 6)

Received: 9 November 2024 Accepted: 29 November 2024 Published: 18 December 2024

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Abstract

In this paper, we consider the nonparametric recursive kernel density estimator on a compact ensemble when observations are censored and β-mixing. In this type of model, it is widely recognized that the traditional empirical distribution does not allow the densities F and G to be efficiently evaluated. Thus, Kaplan and Meier suggested a consistent estimator of Gnto properly estimate G. Let {T_k, k ≥ 1} be a strictly stationary sequence of random variables distributed as T. We aims to establish a strong uniform consistency on a compact set with a rate of recursive kernel estimator of the underlying density function f when the random variable of interest T is right censored by another C variable. In censoring, the observation is only partially known, which means that there are only the n pairs (Y_i, δ_i), Y_i= min(T_i, C_i) and δ_i= II_{{T_i≤C_i}}, where II_A, where the indicator function for event A. Firstly, we propose the uniform convergence of this recursive estimator towards the density f. Then, we showed the veracity of our results by establishing all the necessary proofs. In other words we will prove our main result by establishing three lemmas. And finally we validated our theoretical results with a simulation study.

Published in	American Journal of Theoretical and Applied Statistics (Volume 13, Issue 6)
DOI	10.11648/j.ajtas.20241306.17
Page(s)	255-265
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Censored Data, Kernel Estimator, Density Function, Sure Convergence, β-mixing

References

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[2]	Davies, I. 1973. Strong consistency of a sequential estimator of a probability density function. Bulletin of Mathematical Statistics 15(3-4): 49-54. https://doi.org/10.5109/13071
[3]	Devroye, L. 1979. On the pointwise and integral convergence of recursive kernel estimates of probability densities. Utilitas Math 15: 113-28.
[4]	Diehl, S. and Stute, W. Kernel density and hazard function estimation in the presence of censoring..J. Multivariate Anal., 25(1988), no. 2, 299-310.
[5]	Duflo, M. Random iterative models. In Collection Applications of Mathematics; Vol. 34. Springer-Verlag: Berlin, Heidelberg. https://doi.org/10.1007/978-3-662- 12880-0
[6]	Ferrani Yacine 2014. On the non-parametric estimation of density and mode in models incomplete and associated data.
[7]	Fldes, A., Rejto, L. and Winter, B. B. Strong consistency properties of nonparametric estimators for randomly censored data, part II: Estimation of density and failure rate Period. Math. Hungar, 12(1981), no. 1, 15-29.
[8]	Jmaei, A.; Slaoui, Y.; Dellagi, W. Recursive distribution estimators defined by stochastic approximation method using bernstein polynomials. J. Nonparametr. Stat. 2017, 29, 792-805. https://doi.org/10.1080/10485252.2017.1369538
[9]	Kaplan, E. L., Meier, P. (1958). Non parametric estimationfromincompleteobservations.J.Amer.Statist. Assoc. 53: 457-481.
[10]	Kushner, H. J.; Yin, G. G. Stochastic approximation and recursive algorithms and applications. Stoch. Model. Appl. Probab. 2003, 35.
[11]	Mokkadem, A.; Pelletier, M.; Slaoui, Y. The stochastic approximation method for the estimation of a multivariate probability density. J. Statist. Plann. Inference 2009, 139, 2459-2478. https://doi.org/10.1016/j.jspi.2008.11.012
[12]	Noureddine Rhomari. 2001. Approximation and exponential inequalities for are dependent random vectors.
[13]	Robbins, H. and Monro, S. A stochastic approximation method. Ann. Math. Stat., 22(1951), no. 2, 400-407.
[14]	Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U.S.A.
[15]	Parzen, E. On estimation of a probability density and mode. Ann. Math. Statist., 33 (1962), no. 3, 1065-1076.
[16]	Slaoui, Y. Bandwidth selection for recursive kernel density estimators defined by stochastic approximation method. J. Probab. Stat. 2014, 1-11. ID 739640, https://doi.org/10.1155/2014/739640
[17]	Slaoui, Y. 2016a. Optimal bandwidth selection for semi-recursive kernel regression estimators. Statistics and Its Interface 9(3): 375-88. https://doi.org/10.4310/SII.2016.v9.n3.a11
[18]	Slaoui, Yousri (2019) "Smoothing Parameters for Recursive Kernel Density Estimators under Censoring," Communications on Stochastic Analysis: Vol. 13: No. 2, Article 2. https://doi.org/10.31390/cosa.13.2.02
[19]	Volkonskii, V. et Rozanov, Y. (1959). Some limit theorems for random functions. Theor. Probab. Appl., 4: 178.

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APA Style

Mar, M., Diouf, S., Deme, E. H. (2024). Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure. American Journal of Theoretical and Applied Statistics, 13(6), 255-265. https://doi.org/10.11648/j.ajtas.20241306.17

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ACS Style

Mar, M.; Diouf, S.; Deme, E. H. Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure. Am. J. Theor. Appl. Stat. 2024, 13(6), 255-265. doi: 10.11648/j.ajtas.20241306.17

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AMA Style

Mar M, Diouf S, Deme EH. Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure. Am J Theor Appl Stat. 2024;13(6):255-265. doi: 10.11648/j.ajtas.20241306.17

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@article{10.11648/j.ajtas.20241306.17,
  author = {Mouhamed Mar and Saliou Diouf and El Hadji Deme},
  title = {Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {13},
  number = {6},
  pages = {255-265},
  doi = {10.11648/j.ajtas.20241306.17},
  url = {https://doi.org/10.11648/j.ajtas.20241306.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20241306.17},
  abstract = {In this paper, we consider the nonparametric recursive kernel density estimator on a compact ensemble when observations are censored and β-mixing. In this type of model, it is widely recognized that the traditional empirical distribution does not allow the densities F and G to be efficiently evaluated. Thus, Kaplan and Meier suggested a consistent estimator of Gnto properly estimate G. Let {Tk, k ≥ 1} be a strictly stationary sequence of random variables distributed as T. We aims to establish a strong uniform consistency on a compact set with a rate of recursive kernel estimator of the underlying density function f when the random variable of interest T is right censored by another C variable. In censoring, the observation is only partially known, which means that there are only the n pairs (Yi, δi), Yi= min(Ti, Ci) and δi= II{Ti≤Ci}, where IIA, where the indicator function for event A. Firstly, we propose the uniform convergence of this recursive estimator towards the density f. Then, we showed the veracity of our results by establishing all the necessary proofs. In other words we will prove our main result by establishing three lemmas. And finally we validated our theoretical results with a simulation study.},
 year = {2024}
}

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TY  - JOUR
T1  - Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure
AU  - Mouhamed Mar
AU  - Saliou Diouf
AU  - El Hadji Deme
Y1  - 2024/12/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ajtas.20241306.17
DO  - 10.11648/j.ajtas.20241306.17
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 255
EP  - 265
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20241306.17
AB  - In this paper, we consider the nonparametric recursive kernel density estimator on a compact ensemble when observations are censored and β-mixing. In this type of model, it is widely recognized that the traditional empirical distribution does not allow the densities F and G to be efficiently evaluated. Thus, Kaplan and Meier suggested a consistent estimator of Gnto properly estimate G. Let {Tk, k ≥ 1} be a strictly stationary sequence of random variables distributed as T. We aims to establish a strong uniform consistency on a compact set with a rate of recursive kernel estimator of the underlying density function f when the random variable of interest T is right censored by another C variable. In censoring, the observation is only partially known, which means that there are only the n pairs (Yi, δi), Yi= min(Ti, Ci) and δi= II{Ti≤Ci}, where IIA, where the indicator function for event A. Firstly, we propose the uniform convergence of this recursive estimator towards the density f. Then, we showed the veracity of our results by establishing all the necessary proofs. In other words we will prove our main result by establishing three lemmas. And finally we validated our theoretical results with a simulation study.
VL  - 13
IS  - 6
ER  -

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Author Information

Mouhamed Mar

Laboratory for Study in Statistics and Development, (LERSTAD), Gaston Berger University, Saint-Louis, Senegal

Contact Email

http://orcid.org/0009-0002-0073-7654
Saliou Diouf

Laboratory for Study in Statistics and Development, (LERSTAD), Gaston Berger University, Saint-Louis, Senegal

Contact Email

http://orcid.org/0009-0003-8612-2315
El Hadji Deme

Laboratory for Study in Statistics and Development, (LERSTAD), Gaston Berger University, Saint-Louis, Senegal

Contact Email

http://orcid.org/0009-0004-4277-6779

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APA Style

Mar, M., Diouf, S., Deme, E. H. (2024). Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure. American Journal of Theoretical and Applied Statistics, 13(6), 255-265. https://doi.org/10.11648/j.ajtas.20241306.17

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ACS Style

Mar, M.; Diouf, S.; Deme, E. H. Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure. Am. J. Theor. Appl. Stat. 2024, 13(6), 255-265. doi: 10.11648/j.ajtas.20241306.17

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AMA Style

Mar M, Diouf S, Deme EH. Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure. Am J Theor Appl Stat. 2024;13(6):255-265. doi: 10.11648/j.ajtas.20241306.17

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@article{10.11648/j.ajtas.20241306.17,
  author = {Mouhamed Mar and Saliou Diouf and El Hadji Deme},
  title = {Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {13},
  number = {6},
  pages = {255-265},
  doi = {10.11648/j.ajtas.20241306.17},
  url = {https://doi.org/10.11648/j.ajtas.20241306.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20241306.17},
  abstract = {In this paper, we consider the nonparametric recursive kernel density estimator on a compact ensemble when observations are censored and β-mixing. In this type of model, it is widely recognized that the traditional empirical distribution does not allow the densities F and G to be efficiently evaluated. Thus, Kaplan and Meier suggested a consistent estimator of Gnto properly estimate G. Let {Tk, k ≥ 1} be a strictly stationary sequence of random variables distributed as T. We aims to establish a strong uniform consistency on a compact set with a rate of recursive kernel estimator of the underlying density function f when the random variable of interest T is right censored by another C variable. In censoring, the observation is only partially known, which means that there are only the n pairs (Yi, δi), Yi= min(Ti, Ci) and δi= II{Ti≤Ci}, where IIA, where the indicator function for event A. Firstly, we propose the uniform convergence of this recursive estimator towards the density f. Then, we showed the veracity of our results by establishing all the necessary proofs. In other words we will prove our main result by establishing three lemmas. And finally we validated our theoretical results with a simulation study.},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Recursive Kernel Density Estimators Under Censoring Verifying an β-mixing Dependence Structure
AU  - Mouhamed Mar
AU  - Saliou Diouf
AU  - El Hadji Deme
Y1  - 2024/12/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ajtas.20241306.17
DO  - 10.11648/j.ajtas.20241306.17
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 255
EP  - 265
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20241306.17
AB  - In this paper, we consider the nonparametric recursive kernel density estimator on a compact ensemble when observations are censored and β-mixing. In this type of model, it is widely recognized that the traditional empirical distribution does not allow the densities F and G to be efficiently evaluated. Thus, Kaplan and Meier suggested a consistent estimator of Gnto properly estimate G. Let {Tk, k ≥ 1} be a strictly stationary sequence of random variables distributed as T. We aims to establish a strong uniform consistency on a compact set with a rate of recursive kernel estimator of the underlying density function f when the random variable of interest T is right censored by another C variable. In censoring, the observation is only partially known, which means that there are only the n pairs (Yi, δi), Yi= min(Ti, Ci) and δi= II{Ti≤Ci}, where IIA, where the indicator function for event A. Firstly, we propose the uniform convergence of this recursive estimator towards the density f. Then, we showed the veracity of our results by establishing all the necessary proofs. In other words we will prove our main result by establishing three lemmas. And finally we validated our theoretical results with a simulation study.
VL  - 13
IS  - 6
ER  -

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