Some Asymptotic Properties of a Kernel Bispectum Estimate with Different Multitapers

Mahmoud El-Morshedy (), Abd El-Moneim A. M. Teamah, Mohammed H. El-Menshawy, Rashad M. EL-Sagheer, Hasnaa M. Faied, Afrah Al-Bossly and Mohamed S. Eliwa
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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
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
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

Mathematics, 2022, vol. 10, issue 18, 1-23

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.

Keywords: bispectrum estimate; stationary time series; overlapping segments; multitapering; kernels; bandwidth; DCGINAR(1) process (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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