{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,9]],"date-time":"2026-11-09T22:42:20Z","timestamp":1794264140022,"version":"4.9.0"},"reference-count":32,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T00:00:00Z","timestamp":1622073600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:p>The inverse of a noncentral Wishart matrix occurs in a variety of contexts in multivariate statistical work, including instrumental variable (IV) regression, but there has been very little work on its properties. In this paper, we first provide an expression for the expectation of the inverse of a noncentral Wishart matrix, and then go on to do the same for a number of scalar-valued functions of the inverse. The main result is obtained by exploiting simple but powerful group-equivariance properties of the expectation map involved. Subsequent results exploit the consequences of other invariance properties.<\/jats:p>","DOI":"10.1017\/s0266466621000190","type":"journal-article","created":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T11:15:37Z","timestamp":1622114137000},"page":"1092-1116","source":"Crossref","is-referenced-by-count":10,"title":["PROPERTIES OF THE INVERSE OF A NONCENTRAL WISHART MATRIX"],"prefix":"10.1017","volume":"38","author":[{"given":"Grant","family":"Hillier","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Raymond","family":"Kan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2021,5,27]]},"reference":[{"key":"S0266466621000190_r3","doi-asserted-by":"publisher","DOI":"10.1007\/s10801-018-0846-y"},{"key":"S0266466621000190_r15","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177698153"},{"key":"S0266466621000190_r24","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534891.001.0001","volume-title":"Symmetric Functions and Hall Polynomials","author":"Macdonald","year":"1995"},{"key":"S0266466621000190_r17","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177699530"},{"key":"S0266466621000190_r32","doi-asserted-by":"crossref","DOI":"10.1214\/lnms\/1215465457","volume-title":"A Statistical Approach to Zonal Polynomials","volume":"4","author":"Takemura","year":"1984"},{"key":"S0266466621000190_r9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02595828"},{"key":"S0266466621000190_r6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02480302"},{"key":"S0266466621000190_r13","doi-asserted-by":"publisher","DOI":"10.2307\/1911467"},{"key":"S0266466621000190_r2","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(74)90014-1"},{"key":"S0266466621000190_r14","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177704980"},{"key":"S0266466621000190_r22","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(84)90038-1"},{"key":"S0266466621000190_r7","volume-title":"Multivariate Analysis-V","author":"Davis","year":"1980"},{"key":"S0266466621000190_r20","doi-asserted-by":"publisher","DOI":"10.1137\/0519049"},{"key":"S0266466621000190_r27","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316559"},{"key":"S0266466621000190_r16","doi-asserted-by":"publisher","DOI":"10.1007\/s11786-020-00458-0"},{"key":"S0266466621000190_r29","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600012408"},{"key":"S0266466621000190_r23","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2008.04.006"},{"key":"S0266466621000190_r33","first-page":"49","article-title":"On the inverse moments of the non central Wishart matrix","volume":"1","author":"Ullah","year":"1994","journal-title":"Parisankhyan Samikkha"},{"key":"S0266466621000190_r1","first-page":"122","volume-title":"Advances in Economics and Econometrics, Theory and Applications: Ninth World Congress of the Econometric Society","volume":"III","author":"Andrews","year":"2005"},{"key":"S0266466621000190_r28","volume-title":"Handbook of Econometrics","volume":"I","author":"Phillips","year":"1983"},{"key":"S0266466621000190_r4","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177703863"},{"key":"S0266466621000190_r26","doi-asserted-by":"publisher","DOI":"10.1186\/s40064-015-1506-5"},{"key":"S0266466621000190_r31","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(89)90015-7"},{"key":"S0266466621000190_r19","first-page":"278","article-title":"The moments of a doubly non-central t-distribution","volume":"62","author":"Krishnan","year":"1967","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466621000190_r21","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(81)90085-3"},{"key":"S0266466621000190_r30","doi-asserted-by":"publisher","DOI":"10.1007\/BF02481012"},{"key":"S0266466621000190_r25","unstructured":"Macdonald, I. 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