{"id":1579,"date":"2019-10-01T13:29:00","date_gmt":"2019-10-01T13:29:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1579"},"modified":"2026-01-13T18:12:26","modified_gmt":"2026-01-13T18:12:26","slug":"desirable-properties-estimator","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/desirable-properties-estimator\/","title":{"rendered":"Desirable Properties of an Estimator"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Sampling and Estimation (2021 Level I CFA\u00ae Exam \u2013 Reading 10)\",\n  \"description\": \"This CFA Level I Quantitative Methods video explains sampling and estimation fundamentals. It covers simple and stratified sampling, sampling errors, central limit theorem, standard error, confidence intervals, Student\u2019s t-distribution, and biases like survivorship and look-ahead. Key statistical terms and formulas are explained with examples relevant to financial analysts and investment professionals.\",\n  \"uploadDate\": \"2018-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/i.ytimg.com\/vi\/mgY_3CHHYBw\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=mgY_3CHHYBw\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/mgY_3CHHYBw\",\n  \"duration\": \"PT32M00S\"\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/mgY_3CHHYBw?si=K7FcZB_BItk-P1NX\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\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; and<\/li>\n<li>consistency.<\/li>\n<\/ol>\n<p>Let\u2019s now look at each property in detail.<\/p>\n<h2><strong>Unbiasedness<\/strong><\/h2>\n<p>We say that the 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>\u00a0\u00a0 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. As such, this difference is and should be zero if an estimator is unbiased. Otherwise, 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\u00a0<\/sub>and \u03b2&#8217;<sub>j2<\/sub> \u2013 of the population parameter, \u03b2<sub>j<\/sub>:<\/p>\n<p>E(\u03b2&#8217;<sub>j1<\/sub>) = \u03b2<sub>j\u00ad<\/sub>\u00a0 and E(\u03b2&#8217;<sub>j2<\/sub>) = \u03b2<sub>j\u00ad<\/sub><\/p>\n<p>We say that \u03b2&#8217;<sub>j1\u00a0<\/sub>is more efficient relative to \u03b2&#8217;<sub>j2 \u00a0<\/sub>if the variance of the sample distribution of \u03b2&#8217;<sub>j1\u00a0<\/sub>is less than that of \u03b2&#8217;<sub>j2\u00a0<\/sub> for all finite sample sizes.<\/p>\n<p>In short, if we have two unbiased estimators, we should give preference to the estimator with a smaller variance because this means it\u2019s more precise in statistical terms. It\u2019s also important to note that the property of efficiency only applies in the presence of unbiasedness since 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 N 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>or <strong>collapses on<\/strong> the true value of the population parameter \u03b2<sub>j\u00ad<\/sub> as N tends to infinity.<\/p>\n<p>This intuitively means that if a PE\u00a0 is consistent, its distribution becomes more and more concentrated around the real value of the population parameter involved. Therefore, we could say that as N increases, the probability that the estimator \u2018closes in\u2019 on the actual value of the parameter approaches 1.<\/p>\n<p><em>Reading 10 LOS 10g:<\/em><\/p>\n<p><em>Identify and describe desirable properties of an estimator.<\/em><\/p>\n<div class=\"notes_inv\"><hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>","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":2,"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-1579","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 v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Properties of an Estimator | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Understand key properties of estimators, including unbiasedness, efficiency, and consistency, in statistical analysis.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/desirable-properties-estimator\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Properties of an Estimator | CFA Level 1 - 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