{"id":57,"date":"2018-12-24T05:42:57","date_gmt":"2018-12-24T05:42:57","guid":{"rendered":"http:\/\/statp05.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=57"},"modified":"2022-01-07T09:02:15","modified_gmt":"2022-01-07T09:02:15","slug":"small-sample-properties-of-u-statistics","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/chapter\/small-sample-properties-of-u-statistics\/","title":{"rendered":"Small 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 A few concepts<\/strong>\r\n\r\n \r\n

Suppose X = (X1; X2; ::; Xn) are iid observations from an absolutely continuous distribution F . F is the class of all absolutely continuous DFs. Y = (X(1); X(2); ::; X(n)) is the full set of order statistic corresponding to X. Pn is the set of n! permutations of the rst n natural numbers. The joint pdf of (X1; X2; ::; Xn) given the full set of order statistics is<\/p>\r\n\"\"\r\n\r\n4 U Statistic & MVUE<\/strong>\r\n\r\n \r\n

Suppose F is the class of all absolutely continuous DFs. Then Y = (X(1); X(2); ::; X(n)) is complete su cient for F(see, Fraser,1957). Un is symmetric in the observations and hence is a function of the full set of order statistics. Thus Un is Unbiased and function of complete su cient statistics. Therefore Un is MVUE for F<\/p>\r\n \r\n\r\n5 Exact variance of U Statistic<\/strong>\r\n\r\n \r\n\r\n\"\"\r\n\r\n6 Derivation of V ar(Un)<\/strong>\r\n\r\n \r\n

First of all note that Un is a sum of dependent variables. Therefore the covariance terms are not always zero. Thus<\/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\r\n
you can view video on Small Sample Properties of U Statistics<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"
\"epgp<\/a>
\n<\/span><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

1 A few concepts<\/strong><\/p>\n

 <\/p>\n

Suppose X = (X1; X2; ::; Xn) are iid observations from an absolutely continuous distribution F . F is the class of all absolutely continuous DFs. Y = (X(1); X(2); ::; X(n)) is the full set of order statistic corresponding to X. Pn is the set of n! permutations of the rst n natural numbers. The joint pdf of (X1; X2; ::; Xn) given the full set of order statistics is<\/p>\n

\"\"<\/p>\n

4 U Statistic & MVUE<\/strong><\/p>\n

 <\/p>\n

Suppose F is the class of all absolutely continuous DFs. Then Y = (X(1); X(2); ::; X(n)) is complete su cient for F(see, Fraser,1957). Un is symmetric in the observations and hence is a function of the full set of order statistics. Thus Un is Unbiased and function of complete su cient statistics. Therefore Un is MVUE for F<\/p>\n

 <\/p>\n

5 Exact variance of U Statistic<\/strong><\/p>\n

 <\/p>\n

\"\"<\/p>\n

6 Derivation of V ar(Un)<\/strong><\/p>\n

 <\/p>\n

First of all note that Un is a sum of dependent variables. Therefore the covariance terms are not always zero. Thus<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

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 <\/p>\n\n\n\n
you can view video on Small Sample Properties of U Statistics<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":3,"menu_order":4,"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-57","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\/57","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":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapters\/57\/revisions"}],"predecessor-version":[{"id":458,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapters\/57\/revisions\/458"}],"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\/57\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/wp\/v2\/media?parent=57"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/pressbooks\/v2\/chapter-type?post=57"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/wp\/v2\/contributor?post=57"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp05\/wp-json\/wp\/v2\/license?post=57"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}