On the rate of convergence of degenerate U-statistics
Let X, X1, X2, ... be independent identically distributed random variables taking values in a measurable space (Ω, ℜ). Let h(x, y) be real valued measurable symmetric function of the arguments x, y ∈ ℜ. Assume that Eh(x, X)= 0, for all x. We consider U-statistics of type T = n−1 ∑1 ≤ i< k ≤ n h...
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2005-12-01
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| Series: | Lietuvos Matematikos Rinkinys |
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| Online Access: | https://www.journals.vu.lt/LMR/article/view/29319 |
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| _version_ | 1832593206039019520 |
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| author | Olga Januškevičienė |
| author_facet | Olga Januškevičienė |
| author_sort | Olga Januškevičienė |
| collection | DOAJ |
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Let X, X1, X2, ... be independent identically distributed random variables taking values in a measurable space (Ω, ℜ). Let h(x, y) be real valued measurable symmetric function of the arguments x, y ∈ ℜ.
Assume that Eh(x, X)= 0, for all x. We consider U-statistics of type T = n−1 ∑1 ≤ i< k ≤ n h(Xi, Xk). Let qi, i ≥ 1 be eigenvalues of the Hilbert-Schmidt operator associated with the kernel h(x, y) and q1 be the largest eigenvalue. Under the condition β3 := E|h(X, X1)|3 <∞, we prove that Δn = ρ(T, T0) ≤ cβ3q−11 n−1/7 +cq−11∑i ≥ 1 qin−1/4, where T0 is the limit statistic and ρ is a Kolmogorov (or uniform) distance.
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| format | Article |
| id | doaj-art-8b78ae60140b4f5da6dd460906c7c231 |
| institution | Kabale University |
| issn | 0132-2818 2335-898X |
| language | English |
| publishDate | 2005-12-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Lietuvos Matematikos Rinkinys |
| spelling | doaj-art-8b78ae60140b4f5da6dd460906c7c2312025-01-20T18:15:39ZengVilnius University PressLietuvos Matematikos Rinkinys0132-28182335-898X2005-12-0145spec.10.15388/LMR.2005.29319On the rate of convergence of degenerate U-statisticsOlga Januškevičienė0Institute of Mathematics and Informatics Let X, X1, X2, ... be independent identically distributed random variables taking values in a measurable space (Ω, ℜ). Let h(x, y) be real valued measurable symmetric function of the arguments x, y ∈ ℜ. Assume that Eh(x, X)= 0, for all x. We consider U-statistics of type T = n−1 ∑1 ≤ i< k ≤ n h(Xi, Xk). Let qi, i ≥ 1 be eigenvalues of the Hilbert-Schmidt operator associated with the kernel h(x, y) and q1 be the largest eigenvalue. Under the condition β3 := E|h(X, X1)|3 <∞, we prove that Δn = ρ(T, T0) ≤ cβ3q−11 n−1/7 +cq−11∑i ≥ 1 qin−1/4, where T0 is the limit statistic and ρ is a Kolmogorov (or uniform) distance. https://www.journals.vu.lt/LMR/article/view/29319U-statistics of second degreerate of convergencedegenerate U-statistics |
| spellingShingle | Olga Januškevičienė On the rate of convergence of degenerate U-statistics Lietuvos Matematikos Rinkinys U-statistics of second degree rate of convergence degenerate U-statistics |
| title | On the rate of convergence of degenerate U-statistics |
| title_full | On the rate of convergence of degenerate U-statistics |
| title_fullStr | On the rate of convergence of degenerate U-statistics |
| title_full_unstemmed | On the rate of convergence of degenerate U-statistics |
| title_short | On the rate of convergence of degenerate U-statistics |
| title_sort | on the rate of convergence of degenerate u statistics |
| topic | U-statistics of second degree rate of convergence degenerate U-statistics |
| url | https://www.journals.vu.lt/LMR/article/view/29319 |
| work_keys_str_mv | AT olgajanuskeviciene ontherateofconvergenceofdegenerateustatistics |