Discrete approximation of integral operators. (English) Zbl 1119.47067
Summary: A method to approximate the eigenvalues of linear operators depending on an unknown distribution is introduced and applied to weighted sums of squared normally distributed random variables. This area of application includes the approximation of the asymptotic null distribution of certain degenerated U- and V-statistics.
MSC:
| 47N30 | Applications of operator theory in probability theory and statistics |
| 47G10 | Integral operators |
| 47A75 | Eigenvalue problems for linear operators |
| 62E20 | Asymptotic distribution theory in statistics |
| 62G20 | Asymptotic properties of nonparametric inference |
| 65R20 | Numerical methods for integral equations |
| 65J05 | General theory of numerical analysis in abstract spaces |
| 45C05 | Eigenvalue problems for integral equations |
| 45P05 | Integral operators |
References:
| [1] | L. Baringhaus and C. Franz, On a new multivariate two-sample test, J. Multivariate Anal. 88 (2004), no. 1, 190 – 206. · Zbl 1035.62052 · doi:10.1016/S0047-259X(03)00079-4 |
| [2] | -, A family of multivariate two-sample tests. submitted. |
| [3] | Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. · Zbl 0635.47002 |
| [4] | C. Franz, “Asymptotische Eigenschaften einer neuen Klasse von nichtparametrischen Zwei-Stichproben-Tests.” Doctoral thesis, University of Hannover, 2004. · Zbl 1275.62022 |
| [5] | Vladimir Koltchinskii and Evarist Giné, Random matrix approximation of spectra of integral operators, Bernoulli 6 (2000), no. 1, 113 – 167. · Zbl 0949.60078 · doi:10.2307/3318636 |
| [6] | Georg Neuhaus, Functional limit theorems for \?-statistics in the degenerate case, J. Multivariate Anal. 7 (1977), no. 3, 424 – 439. · Zbl 0368.60034 · doi:10.1016/0047-259X(77)90083-5 |
| [7] | Friedrich Stummel, Diskrete Konvergenz linearer Operatoren. I, Math. Ann. 190 (1970/71), 45 – 92 (German). · Zbl 0203.45301 · doi:10.1007/BF01349967 |
| [8] | Friedrich Stummel, Diskrete Konvergenz linearer Operatoren. II, Math. Z. 120 (1971), 231 – 264 (German). · Zbl 0209.15502 · doi:10.1007/BF01117498 |
| [9] | Friedrich Stummel, Diskrete Konvergenz linearer Operatoren. III, Linear operators and approximation (Proc. Conf., Oberwolfach, 1971), Birkhäuser, Basel, 1972, pp. 196 – 216. Internat. Ser. Numer. Math., Vol. 20 (German). · Zbl 0255.47028 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
