{"id":35430,"date":"2018-02-20T23:50:30","date_gmt":"2018-02-20T23:50:30","guid":{"rendered":"https:\/\/writemyessayfree.com\/a-derive-the-mle-for-%ce%b2-and-%cf%89-show-that-they-indeed-maximize-the-likelihood-b-compute-the-information-matrix-for-%ce%b2-and-vech%cf%89-c-compute-the-asymptotic-variance-of-mles-a"},"modified":"2017-08-16T06:32:43","modified_gmt":"2017-08-16T06:32:43","slug":"a-derive-the-mle-for-%ce%b2-and-%cf%89-show-that-they-indeed-maximize-the-likelihood-b-compute-the-information-matrix-for-%ce%b2-and-vech%cf%89-c-compute-the-asymptotic-variance-of-mles-a","status":"publish","type":"post","link":"https:\/\/www.benedictsol.com\/blogs\/a-derive-the-mle-for-%ce%b2-and-%cf%89-show-that-they-indeed-maximize-the-likelihood-b-compute-the-information-matrix-for-%ce%b2-and-vech%cf%89-c-compute-the-asymptotic-variance-of-mles-a\/","title":{"rendered":"(a) Derive the MLE for \u03b2 and \u03a9. Show that they indeed maximize the likelihood. (b) Compute the information matrix for \u03b2 and vech(\u03a9). (c) Compute the asymptotic variance of MLEs (a) Derive the MLE for \u03b2 and \u03a9. Show that they indeed maximize the likelihood. (b) Compute the information matrix for \u03b2 and vech(\u03a9). (c) Compute the asymptotic variance of MLEs"},"content":{"rendered":"<div class=\"the_content_wrapper\">\n<p>AECO621 : Problem Set 1Due in class on Feb 23, 2017 1. Suppose Y \u2208Rn\u00d7p, X \u2208Rn\u00d7m and B \u2208Rm\u00d7p. Show that (Y \u2212XB)0(Y \u2212XB)\u2265L Y 0MXY, where\u2265L denotesL\u00a8ownerpartialordering, MX = I\u2212PX = I\u2212X<\/p>\n<p>(X0X)\u2212X0, andthattheequalityholdsifandonlyif B takesthevalue B\u2217 = (X0X)\u2212X0Y . 2. Suppose x\u2208Rp, y \u2208Rq are random vectors and that z =? x y ?with cov(z) =? \u03a3xx \u03a3xy \u03a3yx <\/p>\n<p>\u03a3yy ? Find F \u2208Rq\u00d7p that minimizes cov(z\u2212Fx). 3. Suppose we have a linear regression model yi = Xi\u03b2 + \u03b5i (i = 1,\u2219\u2219\u2219 ,n), where \u03b2 is k \u00d71 vector of parameters and Xi are <\/p>\n<p>iid with finite second moment. Assume also \u03b5i|Xi i.i.d. \u223c N(0,\u03a9) with \u03a9 positive definite. (a) Derive the MLE for \u03b2 and \u03a9. Show that they indeed maximize the <\/p>\n<p>likelihood. (b) Compute the information matrix for \u03b2 and vech(\u03a9). (c) Compute the asymptotic variance of MLEs <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>AECO621 : Problem Set 1Due in class on Feb 23, 2017 1. Suppose Y \u2208Rn\u00d7p, X \u2208Rn\u00d7m and B \u2208Rm\u00d7p. Show that (Y \u2212XB)0(Y \u2212XB)\u2265L Y 0MXY, where\u2265L denotesL\u00a8ownerpartialordering, MX = I\u2212PX = I\u2212X (X0X)\u2212X0, andthattheequalityholdsifandonlyif B takesthevalue B\u2217 = <a href=\"https:\/\/www.benedictsol.com\/blogs\/a-derive-the-mle-for-%ce%b2-and-%cf%89-show-that-they-indeed-maximize-the-likelihood-b-compute-the-information-matrix-for-%ce%b2-and-vech%cf%89-c-compute-the-asymptotic-variance-of-mles-a\/\" class=\"read-more\">Read More &#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-35430","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/35430","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/comments?post=35430"}],"version-history":[{"count":0,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/35430\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/media?parent=35430"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/categories?post=35430"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/tags?post=35430"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}