{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,12]],"date-time":"2026-09-12T06:32:38Z","timestamp":1789194758567,"version":"build-2803163510"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2001,2,7]],"date-time":"2001-02-07T00:00:00Z","timestamp":981504000000},"content-version":"unspecified","delay-in-days":6,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2001,2]]},"abstract":"<jats:p>There are many instances in the statistical literature \nin which inference is based on a normalized quadratic form \nin a standard normal vector, normalized by the squared \nlength of that vector. Examples include both test statistics \n(the Durbin\u2013Watson statistic) and estimators (serial \ncorrelation coefficients). Although much studied, no general \nclosed-form expression for the density function of such \na statistic is known. This paper gives general formulae \nfor the density in each open interval between the characteristic \nroots of the matrix involved. Results are given for the \ncase of distinct roots, which need not be assumed positive, \nand when the roots occur with multiplicities greater than \none. Starting from a representation of the density as a surface \nintegral over an (<jats:italic>n<\/jats:italic> \u2212 2)-dimensional hyperplane, \nthe density is expressed in terms of top-order zonal polynomials \ninvolving difference quotients of the characteristic roots of the \nmatrix in the numerator quadratic form.<\/jats:p>","DOI":"10.1017\/s026646660117101x","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T13:45:46Z","timestamp":1027777546000},"page":"1-28","source":"Crossref","is-referenced-by-count":24,"title":["THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED \nON THE <i>n<\/i>-SPHERE"],"prefix":"10.1017","volume":"17","author":[{"given":"Grant","family":"Hillier","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2001,2,7]]},"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S026646660117101X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,9]],"date-time":"2019-04-09T20:10:27Z","timestamp":1554840627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S026646660117101X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,2]]},"references-count":0,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2001,2]]}},"alternative-id":["S026646660117101X"],"URL":"https:\/\/doi.org\/10.1017\/s026646660117101x","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,2]]}}}