{"id":70,"date":"2018-12-24T06:08:11","date_gmt":"2018-12-24T06:08:11","guid":{"rendered":"http:\/\/statp05.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=70"},"modified":"2018-12-24T06:13:53","modified_gmt":"2018-12-24T06:13:53","slug":"large-sample-properties-of-u-statistics","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/chapter\/large-sample-properties-of-u-statistics\/","title":{"rendered":"Large Sample Properties of U statistics"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n1 Large Sample Properties<\/strong>\r\n\r\n \r\n

Unbiasedness and possessing minimum variance are the small sample properties of an esti-mator. To judge the performance of an estimator for large sample sizes, one uses Consistency and asymptotic normality. Consistency gives the limiting value whereas asymptotic normal-ity speci es the rate of convergence to the limiting value. A consistent estimator having asymptotic normality is known as Consistent Asymptotic Normal(CAN). U statistic is a CAN estimator.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

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The rst term in the RHS of (*) converges in distribution to a 2 21 distribution. The second term converges to 2 in probability. Now an application of Slutsky's Theorem gives that the LHS of (*) converges to a 2( 21 1) distribution. Thus we need n as the normalizing factor to get a non-degenerate limiting distribution. However, the limiting distribution under degeneracy is no longer normal.<\/p>\r\n\r\n<\/div>\r\n \r\n\r\n\r\n\r\n
you can view video on Large Sample Properties of U statistics<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"
\"epgp<\/a>
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 <\/p>\n

1 Large Sample Properties<\/strong><\/p>\n

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Unbiasedness and possessing minimum variance are the small sample properties of an esti-mator. To judge the performance of an estimator for large sample sizes, one uses Consistency and asymptotic normality. Consistency gives the limiting value whereas asymptotic normal-ity speci es the rate of convergence to the limiting value. A consistent estimator having asymptotic normality is known as Consistent Asymptotic Normal(CAN). U statistic is a CAN estimator.<\/p>\n

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The rst term in the RHS of (*) converges in distribution to a 2 21 distribution. The second term converges to 2 in probability. Now an application of Slutsky’s Theorem gives that the LHS of (*) converges to a 2( 21 1) distribution. Thus we need n as the normalizing factor to get a non-degenerate limiting distribution. However, the limiting distribution under degeneracy is no longer normal.<\/p>\n<\/div>\n

 <\/p>\n\n\n\n
you can view video on Large Sample Properties of U statistics<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":3,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["mr-taranga-mukherjee"],"pb_section_license":""},"chapter-type":[],"contributor":[59],"license":[],"class_list":["post-70","chapter","type-chapter","status-publish","hentry","contributor-mr-taranga-mukherjee"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapters\/70","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":6,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapters\/70\/revisions"}],"predecessor-version":[{"id":83,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapters\/70\/revisions\/83"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapters\/70\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/wp\/v2\/media?parent=70"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapter-type?post=70"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/wp\/v2\/contributor?post=70"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/wp\/v2\/license?post=70"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}