{"id":31307,"date":"2021-09-22T06:50:17","date_gmt":"2021-09-22T06:50:17","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=31307"},"modified":"2026-01-10T07:37:06","modified_gmt":"2026-01-10T07:37:06","slug":"properties-of-an-estimator","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/properties-of-an-estimator\/","title":{"rendered":"Properties of an Estimator"},"content":{"rendered":"\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/QDmc4Pa92bs\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A point estimator (PE) is a sample statistic used to estimate an unknown population parameter. It is a random variable and therefore varies from sample to sample. A good example of an estimator is the sample mean, \\(x\\), which helps statisticians estimate the population mean, \u03bc. There are three desirable properties every good estimator should possess. These are:<\/p>\n<p><!--more--><\/p>\n<ol>\n<li>Unbiasedness.<\/li>\n<li>Efficiency.<\/li>\n<li>Consistency.<\/li>\n<\/ol>\n<p>Let us now look at each property in detail.<\/p>\n<div style=\"margin: 0 0 20px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n    style=\"\n      display: inline-block;\n      border: 2px solid #1e63ff;\n      color: #1e63ff;\n      background: #ffffff;\n      padding: 10px 14px;\n      border-radius: 10px;\n      font-weight: 500;\n      line-height: 1.35;\n      text-decoration: none;\n    \"\n  >\n    Want to practice estimator properties like unbiasedness, efficiency, and consistency for CFA Level I? Try AnalystPrep\u2019s free trial now.\n  <\/a>\n<\/div>\n\n<h2><strong>Unbiasedness<\/strong><\/h2>\n<p>We say that a PE \u03b2&#8217;<sub>j<\/sub> is an unbiased estimator of the true population parameter \u03b2<sub>j<\/sub> if the expected value of \u03b2&#8217;<sub>j<\/sub> is equal to the true \u03b2<sub>j<\/sub>. Putting this in standard mathematical notation, an estimator is unbiased if:<\/p>\n<p>E(\u03b2&#8217;<sub>j<\/sub>) = \u03b2<sub>j\u00ad<\/sub>&nbsp;&nbsp; as long as the sample size <em>n<\/em> is finite.<\/p>\n<p>Bias is the difference between the expected value of the estimator and the true value of the parameter. Therefore, this difference is and should be zero if an estimator is unbiased. Given the foregoing&nbsp; statement, a non-zero difference indicates bias. A biased estimator can be less or more than the true parameter, giving rise to positive and negative biases.<\/p>\n<h2><strong>Efficiency<\/strong><\/h2>\n<p>Suppose we have two unbiased estimators \u2013 \u03b2&#8217;<sub>j1&nbsp;<\/sub>and \u03b2&#8217;<sub>j2<\/sub>&nbsp;\u2013 of the population parameter \u03b2<sub>j<\/sub>:<\/p>\n<p>E(\u03b2&#8217;<sub>j1<\/sub>) = \u03b2<sub>j\u00ad<\/sub>&nbsp; and E(\u03b2&#8217;<sub>j2<\/sub>) = \u03b2<sub>j\u00ad<\/sub><\/p>\n<p>We say that \u03b2&#8217;<sub>j1&nbsp;<\/sub>is more efficient relative to \u03b2&#8217;<sub>j2 &nbsp;<\/sub>if the variance of the sample distribution of \u03b2&#8217;<sub>j1&nbsp;<\/sub>is less than that of \u03b2&#8217;<sub>j2&nbsp;<\/sub> for all finite sample sizes.<\/p>\n<p>In short, if we have two unbiased estimators, we prefer the estimator with a smaller variance because this means it is more precise in statistical terms. It is also important to note that the property of efficiency only applies in the presence of unbiasedness. This is because we only consider the variances of unbiased estimators.<\/p>\n<h2><strong>Consistency of an Estimator<br><\/strong><\/h2>\n<p>Let \u03b2&#8217;<sub>j<\/sub>(N) denote an estimator of \u03b2<sub>j\u00ad <\/sub>where <em>n<\/em> represents the sample size. We would consider \u03b2&#8217;<sub>j<\/sub>(N) a consistent point estimator of \u03b2<sub>j\u00ad<\/sub> if its sampling distribution <strong>converges to<\/strong>&nbsp;the true value of the population parameter \u03b2<sub>j\u00ad<\/sub> as <em>n<\/em> tends to infinity.<\/p>\n<p>This intuitively means that if a PE is consistent, its distribution becomes more concentrated around the real value of the population parameter involved. As such, we could say that as <em>n<\/em> increases, the probability that the estimator \u2018closes in\u2019 on the actual value of the parameter approaches 1.<\/p>\n<div style=\"text-align: center; margin: 32px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n    style=\"\n      display: inline-block;\n      background-color: #1e63ff;\n      color: #ffffff;\n      padding: 12px 26px;\n      border-radius: 12px;\n      font-weight: 600;\n      font-size: 16px;\n      text-decoration: none;\n    \"\n  >\n    Start Free Trial \u2192\n  <\/a>\n\n  <div style=\"margin-top: 10px; font-size: 14px; color: #374151;\">\n    Practice CFA Level I quantitative methods questions on estimator bias, variance, efficiency, and consistency with clear solutions.\n  <\/div>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>A point estimator (PE) is a sample statistic used to estimate an unknown population parameter. It is a random variable and therefore varies from sample to sample. A good example of an estimator is the sample mean, \\(x\\), which helps&#8230;<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-31307","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Properties of an Estimator | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn key properties of estimators, including unbiasedness, efficiency, and consistency, in statistical analysis for investment decisions.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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