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Some Asymptotic Properties of a Kernel Bispectum Estimate with Different Multitapers

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  • Mahmoud El-Morshedy

    (Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
    Department of Statistics and Computer Science, Faculty of Science, Mansoura University, Mansoura 35516, Egypt)

  • Abd El-Moneim A. M. Teamah

    (Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt)

  • Mohammed H. El-Menshawy

    (Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt)

  • Rashad M. EL-Sagheer

    (Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt
    High Institute of Computer and Management Information System, First Statement, New Cairo 11865, Egypt)

  • Hasnaa M. Faied

    (Department of Mathematics, Faculty of Science, Al-Azhar University (Girls Branch), Nasr City 11884, Egypt)

  • Afrah Al-Bossly

    (Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia)

  • Mohamed S. Eliwa

    (Department of Statistics and Operation Research, College of Science, Qassim University, P.O. Box 6644, Buraydah 51482, Saudi Arabia
    Department of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy
    Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt)

Abstract

Assume X 1 , X 2 , … , X N are realizations of N observations from a real-valued discrete parameter third-order stationary process X t , t = 0 ± 1 , ± 2 , … , with bispectrum f X X X ( λ 1 , λ 2 ) where “ − π ≤ λ 1 , λ 2 ≤ π ”. Based on the previous assumption, L different multitapered biperiodograms I X X X ( m t ) j ( λ 1 , λ 2 ) ; j = 1 , 2 , … , L on overlapped segments ( X t ( j ) ; 1 ≤ t < N ) can be constructed. Further, the mean and variance of the average of these different multitapered biperiodograms can be expressed as asymptotic expressions. According to different bispectral windows/kernels ( W β ( j ) ( α 1 , α 2 ) , where “ − π ⩽ α 1 , α 2 ⩽ π ” and β is the bandwidth) and I X X X ( m t ) j ( λ 1 , λ 2 ) , the bispectrum f X X X ( λ 1 , λ 2 ) can be estimated. The asymptotic expressions of the first- and second-ordered moments as well as the integrated relative mean squared error (IMSE) of this estimate are derived. Finally, some estimation results based on numerically generated data from the selected process “DCGINAR(1)” are presented and discussed in detail.

Suggested Citation

  • Mahmoud El-Morshedy & Abd El-Moneim A. M. Teamah & Mohammed H. El-Menshawy & Rashad M. EL-Sagheer & Hasnaa M. Faied & Afrah Al-Bossly & Mohamed S. Eliwa, 2022. "Some Asymptotic Properties of a Kernel Bispectum Estimate with Different Multitapers," Mathematics, MDPI, vol. 10(18), pages 1-23, September.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:18:p:3284-:d:911543
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    References listed on IDEAS

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    1. Chen Zhao‐Guo & E. J. Hannan, 1980. "The Distribution Of Periodogram Ordinates," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 73-82, January.
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