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Open AccessOpen Access JUSTC Mathematics 22 March 2022

Kernel density estimation under widely orthant dependence

  • 1.

    School of Big Data and Artificial Intelligence, Chizhou University, Chizhou 247000, China

  • 2.

    School of Mathematics and Statistics, Chaohu University, Hefei 238024, China

Funds:  Supported by the Provincial Natural Science Research Project of Anhui Colleges (KJ2020A0679) and the Key Research Project of Chizhou University (CZ2019ZRZ003, CZ2020ZRZ06)
Cite this:
https://doi.org/10.52396/JUSTC-2021-0136
More Information
  • Author Bio:

    Wei Wang is a lecturer at School of the Big Data and Artificial Intelligence of Chizhou University. He received his Master's degree from Shanghai Maritime University in 2012. His research interests mainly focus on probability limit theory, parametric and nonparametric statistics

    Xufei Tang is a lecturer at Chaohu University. He received his Master's degree from Anhui University in 2018. His research interests focus on probability limit theory of dependent variables

  • Corresponding author: E-mail: xufei_tang@163.com
  • Received Date: 20 May 2021
  • Accepted Date: 19 December 2021
    • Available Online: 22 March 2022
    Abstract

  • Abstract

    The kernel density estimator for widely orthant dependent random variables is studied. The exponential inequalities and the exponential rate for the estimator of a density function with a uniform version over compact sets are investigated. Further, the consistency of the estimator is proved. The results are generalizations of some existing outcomes for both associated and negatively associated samples. The convergence rate of the kernel density estimator is illustrated via a simulation study. Moreover, a real data analysis is presented.

    Graphical abstract

      We investigate the kernel density estimator for widely orthant dependent random variables and obtain the exponential inequalities and the exponential rate for the estimator of density function with a uniform version over compact sets. The consistency of the estimator is also proved.

    Abstract

    The kernel density estimator for widely orthant dependent random variables is studied. The exponential inequalities and the exponential rate for the estimator of a density function with a uniform version over compact sets are investigated. Further, the consistency of the estimator is proved. The results are generalizations of some existing outcomes for both associated and negatively associated samples. The convergence rate of the kernel density estimator is illustrated via a simulation study. Moreover, a real data analysis is presented.

    • Public Summary

    • We consider samples that satisfy the notion of WOD, which is the most general one among the dependent random variables.
    • The obtained results extend and complement the results of Kheyri et al.[14] to a much more general case.
    • Some numerical analysis and a real data analysis are presented to support the theoretical results.

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    • References(20)
  • [1]
    Parzen E. On estimation of a probability density function and mode. Ann. Math. Stat., 1962, 33 (3): 1065–1076. doi: 10.1214/aoms/1177704472
    [2]
    Rosenblatt M. Curve estimates. Ann. Math. Stat., 1971, 42 (6): 1815–1842. doi: 10.1214/aoms/1177693050
    [3]
    Silverman B W. Density Estimation for Statistics and Data Analysis. London: Chapman and Hall, 1986.
    [4]
    Yakowitz S. Nonparametric density and regression estimation for Markov sequences without mixing assumptions. J. Multivar. Anal., 1989, 30 (1): 124–136. doi: 10.1016/0047-259X(89)90091-2
    [5]
    Bagia I, Rao B. Estimation of the survival functions for stationary associated processes. Stat. Probab. Lett., 1991, 12 (5): 291–385. doi: 10.1016/0167-7152(91)90027-O
    [6]
    Bagia I, Rao B. Kernel-type density and failure rate estimation for associated sequences. Ann. Inst. Stat. Math., 1995, 47 (2): 253–266. doi: 10.1007/BF00773461
    [7]
    Roussas G G. Kernel estimates under association: Strong uniform consistency. Stat. Probab. Lett., 1991, 12 (5): 393–403. doi: 10.1016/0167-7152(91)90028-P
    [8]
    Masry E. Multivariate probability density estimation for associated processes: Strong consistency and rates. Stat Probab. Lett., 2002, 58 (2): 205–219. doi: 10.1016/S0167-7152(02)00105-0
    [9]
    Oliveira P E. Density estimation for associated sampling: A point process influenced approach. J. Nonparametr. Stat., 2002, 14 (6): 495–509. doi: 10.1080/10485250213904
    [10]
    Henriques C, Oliveira P E. Exponential rates for kernel density estimation under association. Stat. Neerl., 2005, 59 (4): 448–466. doi: 10.1111/j.1467-9574.2005.00302.x
    [11]
    Ling N X, Xu C M, Peng X Z. The consistency of density function kernel estimation under NQD samples. Journal of Hefei University of Technology (Natural Science Edition), 2008, 31 (2): 287–290. doi: 10.3969/j.issn.1003-5060.2008.02.030
    [12]
    Su H L, Tang G Q, Sun G H. The consistency of density kernel estimation under ND samples. Journal of Guilin University of Technology, 2013, 33 (3): 565–568. doi: 10.3969/j.issn.1674-9057.2013.03.030
    [13]
    Wang M, Li K C. Generalized consistency of density kernel estimation. Journal of Hubei University of Arts and Science, 2015, 36 (11): 19–22. doi: 10.3969/j.issn.1009-2854.2015.11.004
    [14]
    Kheyri A, Amini M, Jabbari H, et al. Kernel density estimation under negative superadditive dependence and its application for real data. Journal of Statistical Computation and Simulation, 2019, 89 (10): 2373–2392. doi: 10.1080/00949655.2019.1619738
    [15]
    Wang K Y, Wang Y B, Gao Q W. Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodology and Computing in Applied Probability, 2013, 15: 109–124. doi: 10.1007/s11009-011-9226-y
    [16]
    Joag-Dev K, Proschan F. Negative association of random variables with applications. Annals of Statistics, 1983, 11 (1): 286–295. doi: 10.1214/aos/1176346079
    [17]
    Hu T Z. Negatively superadditive dependence of random variables with applications. Chinese Journal of Applied Probability and Statisties, 2000, 16 (2): 133–144. doi: 10.3969/j.issn.1001-4268.2000.02.003
    [18]
    Wang X J, Xu C, Hu T C, et al. On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST, 2014, 23 (3): 607–629. doi: 10.1007/s11749-014-0365-7
    [19]
    Hoeffding W. Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc., 1963, 58 (301): 13–30. doi: 10.1080/01621459.1963.10500830
    [20]
    Liu L. Precise large deviations for dependent random variables with heavy tails. Stat. Probab. Lett., 2009, 79 (9): 1290–1298. doi: 10.1016/j.spl.2009.02.001
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    Figure  1.  Comparison of $ \hat f_n(x) $ and $ f(x) $ for the fixed bandwidth $ h_n = n^{-1/5} $.

    Figure  2.  Comparison of $ \hat f_n(x) $ and $ f(x) $ for the bandwidth based on CV.

    Figure  3.  Boxplot of $ \hat f_n(x)-f(x) $ for $n = 100,\;200,\;400,\;800$.

    Figure  4.  (a) Urbanization rate yearly series; (b) Autocorrelation function; (c) Partial autocorrelation function.

    Figure  5.  (a) Differences of urbanization rate yearly series; (b) Autocorrelation function; (c) Partial autocorrelation function.

    Figure  6.  Comparison of $ \hat f_n(x) $ and $ f(x) $ for different bandwidths.

    [1]
    Parzen E. On estimation of a probability density function and mode. Ann. Math. Stat., 1962, 33 (3): 1065–1076. doi: 10.1214/aoms/1177704472
    [2]
    Rosenblatt M. Curve estimates. Ann. Math. Stat., 1971, 42 (6): 1815–1842. doi: 10.1214/aoms/1177693050
    [3]
    Silverman B W. Density Estimation for Statistics and Data Analysis. London: Chapman and Hall, 1986.
    [4]
    Yakowitz S. Nonparametric density and regression estimation for Markov sequences without mixing assumptions. J. Multivar. Anal., 1989, 30 (1): 124–136. doi: 10.1016/0047-259X(89)90091-2
    [5]
    Bagia I, Rao B. Estimation of the survival functions for stationary associated processes. Stat. Probab. Lett., 1991, 12 (5): 291–385. doi: 10.1016/0167-7152(91)90027-O
    [6]
    Bagia I, Rao B. Kernel-type density and failure rate estimation for associated sequences. Ann. Inst. Stat. Math., 1995, 47 (2): 253–266. doi: 10.1007/BF00773461
    [7]
    Roussas G G. Kernel estimates under association: Strong uniform consistency. Stat. Probab. Lett., 1991, 12 (5): 393–403. doi: 10.1016/0167-7152(91)90028-P
    [8]
    Masry E. Multivariate probability density estimation for associated processes: Strong consistency and rates. Stat Probab. Lett., 2002, 58 (2): 205–219. doi: 10.1016/S0167-7152(02)00105-0
    [9]
    Oliveira P E. Density estimation for associated sampling: A point process influenced approach. J. Nonparametr. Stat., 2002, 14 (6): 495–509. doi: 10.1080/10485250213904
    [10]
    Henriques C, Oliveira P E. Exponential rates for kernel density estimation under association. Stat. Neerl., 2005, 59 (4): 448–466. doi: 10.1111/j.1467-9574.2005.00302.x
    [11]
    Ling N X, Xu C M, Peng X Z. The consistency of density function kernel estimation under NQD samples. Journal of Hefei University of Technology (Natural Science Edition), 2008, 31 (2): 287–290. doi: 10.3969/j.issn.1003-5060.2008.02.030
    [12]
    Su H L, Tang G Q, Sun G H. The consistency of density kernel estimation under ND samples. Journal of Guilin University of Technology, 2013, 33 (3): 565–568. doi: 10.3969/j.issn.1674-9057.2013.03.030
    [13]
    Wang M, Li K C. Generalized consistency of density kernel estimation. Journal of Hubei University of Arts and Science, 2015, 36 (11): 19–22. doi: 10.3969/j.issn.1009-2854.2015.11.004
    [14]
    Kheyri A, Amini M, Jabbari H, et al. Kernel density estimation under negative superadditive dependence and its application for real data. Journal of Statistical Computation and Simulation, 2019, 89 (10): 2373–2392. doi: 10.1080/00949655.2019.1619738
    [15]
    Wang K Y, Wang Y B, Gao Q W. Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodology and Computing in Applied Probability, 2013, 15: 109–124. doi: 10.1007/s11009-011-9226-y
    [16]
    Joag-Dev K, Proschan F. Negative association of random variables with applications. Annals of Statistics, 1983, 11 (1): 286–295. doi: 10.1214/aos/1176346079
    [17]
    Hu T Z. Negatively superadditive dependence of random variables with applications. Chinese Journal of Applied Probability and Statisties, 2000, 16 (2): 133–144. doi: 10.3969/j.issn.1001-4268.2000.02.003
    [18]
    Wang X J, Xu C, Hu T C, et al. On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST, 2014, 23 (3): 607–629. doi: 10.1007/s11749-014-0365-7
    [19]
    Hoeffding W. Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc., 1963, 58 (301): 13–30. doi: 10.1080/01621459.1963.10500830
    [20]
    Liu L. Precise large deviations for dependent random variables with heavy tails. Stat. Probab. Lett., 2009, 79 (9): 1290–1298. doi: 10.1016/j.spl.2009.02.001
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