{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T06:53:31Z","timestamp":1759042411823},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":4943,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1995,8]]},"abstract":"<jats:p>We obtain simple and generally applicable conditions for the existence of mixed moments <jats:italic>E<\/jats:italic>([<jats:italic>X<\/jats:italic>\u2032 <jats:italic>AX<\/jats:italic>]\u2033\/[<jats:italic>X<\/jats:italic>\u2032<jats:italic>BX<\/jats:italic>]<jats:italic>U<\/jats:italic>) of the ratio of quadratic forms <jats:italic>T<\/jats:italic> = <jats:italic>X<\/jats:italic>\u2032 <jats:italic>AX<\/jats:italic>\/<jats:italic>X<\/jats:italic>\u2032<jats:italic>BX<\/jats:italic> where <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic> are <jats:italic>n<\/jats:italic> \u00d7 <jats:italic>n<\/jats:italic> symmetric matrices and <jats:italic>X<\/jats:italic> is a random <jats:italic>n<\/jats:italic>-vector. Our principal theorem is easily stated when <jats:italic>X<\/jats:italic> has an elliptically symmetric distribution, which class includes the multivariate normal and <jats:italic>t<\/jats:italic> distributions, whether degenerate or not. The result applies to the ratio of multivariate quadratic polynomials and can be expected to remain valid in most situations in which <jats:italic>X<\/jats:italic> is subject to linear constraints.<\/jats:p><jats:p>If <jats:italic>u<\/jats:italic> \u2264 <jats:italic>v<\/jats:italic>, the precise distribution of <jats:italic>X<\/jats:italic>, and in particular the existence of moments of <jats:italic>X<\/jats:italic>, is virtually irrelevant to the existence of the mixed moments of <jats:italic>T<\/jats:italic>; if <jats:italic>u<\/jats:italic> &gt; <jats:italic>v<\/jats:italic>, a prerequisite for existence of the (<jats:italic>u<\/jats:italic>, <jats:italic>v<\/jats:italic>)th mixed moment is the existence of the 2(<jats:italic>u<\/jats:italic> \u2212 <jats:italic>v<\/jats:italic>)th moment of <jats:italic>X<\/jats:italic> When <jats:italic>X<\/jats:italic>is not degenerate, the principal further requirement for the existence of the mixed moment is that <jats:italic>B<\/jats:italic> has rank exceeding 2<jats:italic>v<\/jats:italic>.<\/jats:p>","DOI":"10.1017\/s0266466600009725","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T11:39:43Z","timestamp":1236685183000},"page":"750-774","source":"Crossref","is-referenced-by-count":11,"title":["On the Existence of Moments of Ratios of Quadratic Forms"],"prefix":"10.1017","volume":"11","author":[{"given":"Leigh A.","family":"Roberts","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,2,11]]},"reference":[{"key":"S0266466600009725_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(89)90065-1"},{"key":"S0266466600009725_ref013","volume-title":"Introduction to Topology and Modern Analysis","author":"Simmons","year":"1963"},{"key":"S0266466600009725_ref012","doi-asserted-by":"publisher","DOI":"10.2307\/3214565"},{"key":"S0266466600009725_ref011","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316559"},{"key":"S0266466600009725_ref007","doi-asserted-by":"publisher","DOI":"10.1080\/07474939108800205"},{"key":"S0266466600009725_ref005","volume-title":"Distributions in Statistics: Continuous Univariate Distributions","author":"Johnson","year":"1972"},{"key":"S0266466600009725_ref004","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177728981"},{"key":"S0266466600009725_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-2937-2"},{"key":"S0266466600009725_ref002","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/63.3.661"},{"key":"S0266466600009725_ref001","volume-title":"Mathematical Methods of Statistics","author":"Cramer","year":"1946"},{"key":"S0266466600009725_ref010","volume-title":"An Introduction to Probability Theory","author":"Moran","year":"1968"},{"key":"S0266466600009725_ref009","volume-title":"Quadratic Forms in Random Variables. Statistics: Text-books and Monographs","author":"Mathai","year":"1992"},{"key":"S0266466600009725_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/20075629"},{"key":"S0266466600009725_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(87)90086-1"},{"key":"S0266466600009725_ref015","doi-asserted-by":"publisher","DOI":"10.1080\/03610928908829986"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600009725","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T17:18:12Z","timestamp":1557767892000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600009725\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,8]]},"references-count":15,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1995,8]]}},"alternative-id":["S0266466600009725"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600009725","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,8]]}}}