{"id":76,"date":"2018-05-21T09:31:00","date_gmt":"2018-05-21T07:31:00","guid":{"rendered":"https:\/\/kindsonthegenius.com\/blog\/2018\/05\/21\/theory-of-estimation-unbiased-estimation\/"},"modified":"2020-08-22T08:54:38","modified_gmt":"2020-08-22T06:54:38","slug":"theory-of-estimation-unbiased-estimation","status":"publish","type":"post","link":"https:\/\/kindsonthegenius.com\/blog\/theory-of-estimation-unbiased-estimation\/","title":{"rendered":"Theory of Estimation: Unbiased Estimation"},"content":{"rendered":"<div style=\"text-align: justify;\">I was happy to see that Theory of Estimation was really an easy topic to understand, easier than I always thought.<\/div>\n<div style=\"text-align: justify;\">In this\u00a0 article, I would teach you in very simple way, the theory of estimation and you would understand it very clearly.<\/div>\n<div style=\"text-align: justify;\"><\/div>\n<div style=\"text-align: justify;\">\n<p>The challenge many have sometimes is caused by lecturers not explaining the concept clear enough, especially from the basics.<\/p>\n<p>&nbsp;<\/p>\n<p><b>Content<\/b><\/p>\n<ul>\n<li><a href=\"https:\/\/kindsonthegenius.com\/blog\/theory-of-estimation-unbiased-estimation#t1\">What is Estimation<\/a><\/li>\n<li><a href=\"https:\/\/kindsonthegenius.com\/blog\/theory-of-estimation-unbiased-estimation#t2\">Basic Concepts<\/a><\/li>\n<li><a href=\"https:\/\/kindsonthegenius.com\/blog\/theory-of-estimation-unbiased-estimation#t3\">Unbiased Estimator<\/a><\/li>\n<li><a href=\"https:\/\/kindsonthegenius.com\/blog\/theory-of-estimation-unbiased-estimation#t4\">Sample Mean Example<\/a><\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h3 id=\"t1\">1. Basics: What is Estimation?<\/h3>\n<p>Estimation is the process involved in systematically inferring the hidden or unobserved variable from a given information set using a mathematical mapping between the unkowns and the knowns as well as a criterion for estimation.<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/4.bp.blogspot.com\/-3cazYNKqACY\/WwJ5tDnoy7I\/AAAAAAAAB44\/ne3COmEipi42klKYfJ1-zzbwJHPFlUahACLcBGAs\/s1600\/Theory%2Bof%2BEstimation%2BImage.jpg\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/4.bp.blogspot.com\/-3cazYNKqACY\/WwJ5tDnoy7I\/AAAAAAAAB44\/ne3COmEipi42klKYfJ1-zzbwJHPFlUahACLcBGAs\/s320\/Theory%2Bof%2BEstimation%2BImage.jpg\" width=\"320\" height=\"163\" border=\"0\" data-original-height=\"625\" data-original-width=\"1221\" \/><\/a><\/div>\n<p>To carry out estimation you need the following:<\/p>\n<ul>\n<li><b>Data<\/b>: Set of data<\/li>\n<li><b>Estimator<\/b>: The estimator takes in the data as well as two more items (Objective Function and Model) and then helps us make an estimate.<\/li>\n<li><b>Model<\/b>: This is a mapping between the knowns(your dataset) and the unknowns (the parameter)<\/li>\n<li>Objective Function: This is a mathematical statement the can be mimimized or maximized to find best possible solutions among a set of solutions.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3 id=\"t2\">2. Basic Concepts of Estimation<\/h3>\n<p>Take note of these three basic concepts<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/3.bp.blogspot.com\/-82o-ldJXHeA\/WwJ5ov3ZBdI\/AAAAAAAAB40\/S9m1HlNmZuATad_utfFh8i0CiQS2Xu2vwCLcBGAs\/s1600\/Basic%2BConcepts%2Bof%2BEstimation.jpg\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/3.bp.blogspot.com\/-82o-ldJXHeA\/WwJ5ov3ZBdI\/AAAAAAAAB40\/S9m1HlNmZuATad_utfFh8i0CiQS2Xu2vwCLcBGAs\/s400\/Basic%2BConcepts%2Bof%2BEstimation.jpg\" width=\"400\" height=\"135\" border=\"0\" data-original-height=\"295\" data-original-width=\"856\" \/><\/a><\/div>\n<p>The theory of estimation provides the following to help us in the task of making estimation:<\/p>\n<ul>\n<li>Method for estimating the unknowns (eg. model parameters)<\/li>\n<li>Means for accessing the &#8216;goodness&#8217; of the resulting estimates<\/li>\n<li>Making confident statements about the true values (how sure we are about the estimate)<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3 id=\"t3\">3. Unbiased Estimator<\/h3>\n<p>A statistic could be defined as an unbiased estimate of a given parameter if the mean of hte sampling distribution of that statistic can be proved to be equal to the parameter being estimated.<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/2.bp.blogspot.com\/-fXeyI3khfDU\/WwJ8H0kZCeI\/AAAAAAAAB5I\/KmOZ3VGDOe89PdCQyxMCBEayKc9A-16nQCLcBGAs\/s1600\/Unbiased%2BEstimation.jpg\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/2.bp.blogspot.com\/-fXeyI3khfDU\/WwJ8H0kZCeI\/AAAAAAAAB5I\/KmOZ3VGDOe89PdCQyxMCBEayKc9A-16nQCLcBGAs\/s320\/Unbiased%2BEstimation.jpg\" width=\"320\" height=\"49\" border=\"0\" data-original-height=\"83\" data-original-width=\"514\" \/><\/a><\/div>\n<p>Unbiasedness means, that for a large number of observations(samples), the average over all estimations lies close to the true parameter.<\/p>\n<p>&nbsp;<\/p>\n<h3 id=\"t4\">4. Sample Mean Example<\/h3>\n<p>Given an n-dimensional vector, X<sub>1<\/sub>, . . . ,X<sub>n<\/sub>, prove that the extimator for the means <span style=\"font-family: 'georgia' , 'times new roman' , serif;\">\u03bc<\/span> is unbiased.<\/p>\n<p><b>Solution<\/b><br \/>\nTo estimate the mean, we use the sample mean as an estimator.<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/4.bp.blogspot.com\/-tKHxIspAGXY\/WwKMLSmL0GI\/AAAAAAAAB5c\/0kzNkbSz9nQpcDViTu7sblxb21RmZe6-ACLcBGAs\/s1600\/Sample%2Bmean%2Bestimator.jpg\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/4.bp.blogspot.com\/-tKHxIspAGXY\/WwKMLSmL0GI\/AAAAAAAAB5c\/0kzNkbSz9nQpcDViTu7sblxb21RmZe6-ACLcBGAs\/s200\/Sample%2Bmean%2Bestimator.jpg\" width=\"200\" height=\"103\" border=\"0\" data-original-height=\"112\" data-original-width=\"215\" \/><\/a><\/div>\n<div style=\"clear: both; text-align: center;\"><\/div>\n<p>We now prove that the expected value of the estimator is equal to the true mean <span style=\"font-family: 'georgia' , 'times new roman' , serif;\">\u03bc <\/span>(condition for unbiasedness). This we would do using the linearity of the expected value<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/2.bp.blogspot.com\/--PGNh-Skoz8\/WwKPhQ_tMuI\/AAAAAAAAB5o\/VraXAsAoXfMHjfv2HQVCY-HKimFomej5QCLcBGAs\/s1600\/Proof%2Bof%2BUnbiasedness%2Bof%2BSample%2Bmean.jpg\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/2.bp.blogspot.com\/--PGNh-Skoz8\/WwKPhQ_tMuI\/AAAAAAAAB5o\/VraXAsAoXfMHjfv2HQVCY-HKimFomej5QCLcBGAs\/s400\/Proof%2Bof%2BUnbiasedness%2Bof%2BSample%2Bmean.jpg\" width=\"273\" height=\"400\" border=\"0\" data-original-height=\"558\" data-original-width=\"383\" \/><\/a><\/div>\n<p>From the above, we can conclude that the estimator<\/p>\n<div style=\"clear: both; text-align: center;\"><a style=\"margin-left: 1em; margin-right: 1em;\" href=\"https:\/\/3.bp.blogspot.com\/-3Y_NsLl5DDk\/WwKQSx4al8I\/AAAAAAAAB5w\/4QNZ77dCqoUPX3id-jNZKDaDeAiR7ZbwACLcBGAs\/s1600\/Sample%2Bmean%2Bestimator%2Bsymbol.jpg\"><img decoding=\"async\" src=\"https:\/\/3.bp.blogspot.com\/-3Y_NsLl5DDk\/WwKQSx4al8I\/AAAAAAAAB5w\/4QNZ77dCqoUPX3id-jNZKDaDeAiR7ZbwACLcBGAs\/s1600\/Sample%2Bmean%2Bestimator%2Bsymbol.jpg\" border=\"0\" data-original-height=\"54\" data-original-width=\"68\" \/><\/a><\/div>\n<p>is an unbiased estimator of the sample mean. Thank you for your effort in learning. You can reach me if you find anything difficult.<\/p>\n<p>We consider more examples in the following parts.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I was happy to see that Theory of Estimation was really an easy topic to understand, easier than I always thought. In this\u00a0 article, I &hellip; <\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0},"categories":[552],"tags":[],"_links":{"self":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/76"}],"collection":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/comments?post=76"}],"version-history":[{"count":7,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/76\/revisions"}],"predecessor-version":[{"id":1200,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/76\/revisions\/1200"}],"wp:attachment":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/media?parent=76"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/categories?post=76"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/tags?post=76"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}