{"id":34467,"date":"2022-04-10T00:00:00","date_gmt":"2022-04-10T00:00:00","guid":{"rendered":"https:\/\/nursingstudybay.com\/2022\/04\/10\/statistical-inference-maximum-likelihood-estimation\/"},"modified":"2022-04-10T00:00:00","modified_gmt":"2022-04-10T00:00:00","slug":"statistical-inference-maximum-likelihood-estimation","status":"publish","type":"post","link":"https:\/\/www.homeworkacetutors.com\/assessments\/statistical-inference-maximum-likelihood-estimation\/","title":{"rendered":"Statistical Inference: Maximum Likelihood Estimation"},"content":{"rendered":"<p>Answers to all questions should be handed in to the postbox in the statistics corridor by 2pm on<br \/>\nThursday 25st January. Please remember to fill the assignment form in all the required<br \/>\nparts. You can find the form in the Statistics corridor.<br \/>\n\u2022 Suppose that X\u00af = (X1, . . . , Xn) are i.i.d. continuous random variables with p.d.f. given by<br \/>\nf1,\u03b8(x) = (\u03b8 + 1)x<br \/>\n\u03b81(0,1)(x)<br \/>\nfor \u03b8 &gt; \u22121.<br \/>\n(i) Describe the statistical model. [2 marks]<br \/>\n(ii) Construct the likelihood and plot it for values n = 3 and x = (.2, .6, .1). [4 marks]<br \/>\n(iii) Construct the Maximum Likelihood Estimate \u02c6\u03b8(x1, . . . , xn). [4 marks]<br \/>\n(iv) Compute the mean of the Maximum Likelihood Estimator (MLE). What happens? [2 marks]<br \/>\n(v) Suppose that we do not want to estimate \u03b8 but a functiont of \u03b8 given by<br \/>\n\u03b6(\u03b8) = \u03b8 + 1<br \/>\n\u03b8 + 2<br \/>\n.<br \/>\nWe use the estimator<br \/>\n\u02c6\u03b6(X1, . . . , Xn) = 1<br \/>\nn<br \/>\nXn<br \/>\nx=1<br \/>\nXi<br \/>\n.<br \/>\nIs this an unbiased estimator? Compute the variance and Mean Square Error. [4 marks]<br \/>\n(vi) From Theorem 2 of section 9.2 of the textbook, we can immediately derive that the MLE for<br \/>\nparameter \u03b6 is given by<br \/>\n\u02c6\u03b6MLE(X1, . . . , Xn) =<br \/>\n\u02c6\u03b8(X1, . . . , Xn) + 1<br \/>\n\u02c6\u03b8(X1, . . . , Xn) + 2<br \/>\n= 1 \u2212<br \/>\n1<br \/>\n\u02c6\u03b8(X1, . . . , Xn) + 2<br \/>\n.<br \/>\nCompare the two estimators \u02c6\u03b6 and \u02c6\u03b6MLE, for n = 1. [4 marks]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Answers to all questions should be handed in to the postbox in the statistics corridor by 2pm on Thursday 25st January. Please remember to fill the assignment form [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2769],"tags":[2771,2772,2762,2763],"class_list":["post-34467","post","type-post","status-publish","format-standard","hentry","category-words-doctorate","tag-high-quality-research-paper-writing-services-for-students","tag-best-research-paper-writing-services-in-the-u-s","tag-health-care-assignment-help-and-writing-services","tag-best-essay-writing-service-uk-trustpilot"],"_links":{"self":[{"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/posts\/34467","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/comments?post=34467"}],"version-history":[{"count":0,"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/posts\/34467\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/media?parent=34467"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/categories?post=34467"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.homeworkacetutors.com\/assessments\/wp-json\/wp\/v2\/tags?post=34467"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}