Under general conditions, the asymptotic distribution of degenerate second-order U- and V-statistics is an (infinite) weighted sum of χ2 random variables whose weights are the eigenvalues of an integral operator associated with the kernel of the statistic. Also the behavior of the statistic in terms of power can be characterized through the eigenvalues and the eigenfunctions of the same integral operator. No general algorithm seems to be available to compute these quantities starting from the kernel of the statistic. An algorithm is proposed to approximate (as precisely as needed) the asymptotic distribution of the test statistics, and to build several measures of performance for tests based on U- and V-statistics. The algorithm uses the Wielandt–Nyström method of approximation of an integral operator based on quadrature, and can be used with several methods of numerical integration. An extensive numerical study shows that the Wielandt–Nyström method based on Clenshaw–Curtis quadrature performs very well both for the eigenvalues and the eigenfunctions.

Computing the asymptotic distribution of second-order U- and V-statistics

Seri R.
2022-01-01

Abstract

Under general conditions, the asymptotic distribution of degenerate second-order U- and V-statistics is an (infinite) weighted sum of χ2 random variables whose weights are the eigenvalues of an integral operator associated with the kernel of the statistic. Also the behavior of the statistic in terms of power can be characterized through the eigenvalues and the eigenfunctions of the same integral operator. No general algorithm seems to be available to compute these quantities starting from the kernel of the statistic. An algorithm is proposed to approximate (as precisely as needed) the asymptotic distribution of the test statistics, and to build several measures of performance for tests based on U- and V-statistics. The algorithm uses the Wielandt–Nyström method of approximation of an integral operator based on quadrature, and can be used with several methods of numerical integration. An extensive numerical study shows that the Wielandt–Nyström method based on Clenshaw–Curtis quadrature performs very well both for the eigenvalues and the eigenfunctions.
2022
2022
https://www-sciencedirect-com.insubria.idm.oclc.org/science/article/pii/S0167947322000172
Clenshaw–Curtis quadrature; U- and V-statistics; Wielandt–Nyström method
Seri, R.
File in questo prodotto:
File Dimensione Formato  
J030.pdf

non disponibili

Tipologia: Versione Editoriale (PDF)
Licenza: Copyright dell'editore
Dimensione 1.53 MB
Formato Adobe PDF
1.53 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2137291
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact