{"id":45554,"date":"2023-08-09T06:59:00","date_gmt":"2023-08-09T06:59:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45554"},"modified":"2026-03-24T18:46:32","modified_gmt":"2026-03-24T18:46:32","slug":"resampling-2","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/resampling-2\/","title":{"rendered":"Resampling"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/lm07_img2.jpg\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/lm07_img2.jpg\",\n  \"caption\": \"Bootstrap Resampling Method\",\n  \"width\": 583,\n  \"height\": 439,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Given bootstrap resampling results, what is the estimated standard error of the sample mean?\",\n    \"text\": \"Assume that you are studying the median height of 100 students in a university. You draw a sample of 1000 students and obtain 1000 median heights. The mean across all resample means is 5.8. The sum of squares of the differences between each sample mean and the mean across all resample means is 2.3.\\n\\nThe estimate of the standard error of the sample mean is closest to:\\n\\nA. 0.05.\\n\\nB. 0.08.\\n\\nC. 0.10.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A.\\n\\nThe bootstrap estimate of the standard error is calculated as:\\n\\ns\u0304X = \u221a[ (1 \/ (B \u2212 1)) \u00d7 \u03a3(\u03b8\u0302_b \u2212 \u03b8\u0304)\u00b2 ]\\n\\nWhere:\\nB = 1000\\n\u03a3(\u03b8\u0302_b \u2212 \u03b8\u0304)\u00b2 = 2.3\\n\\nThus:\\n\\ns\u0304X = \u221a(2.3 \/ 999) \u2248 \u221a0.002302 \u2248 0.04798 \u2248 0.05\\n\\nTherefore, the estimated standard error is approximately 0.05.\"\n    }\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/pPHmJGLIbyY\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Resampling refers to the act of repeatedly drawing samples from the original observed data sample for the statistical inference of population parameters. The two commonly used methods of resampling are bootstrap and jackknife.<\/p>\n<h2>Bootstrap<\/h2>\n<p>Using a computer, the bootstrap resampling method simulates drawing multiple random samples from the original sample. Each resample is the same size as the original sample. These resamples are used to create a sampling distribution.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"583\" height=\"439\" class=\"size-full wp-image-45557 aligncenter\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/lm07_img2.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/lm07_img2.jpg 583w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/lm07_img2-300x226.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/lm07_img2-400x301.jpg 400w\" sizes=\"auto, (max-width: 583px) 100vw, 583px\" \/><\/p>\n<p>In the bootstrap method, the number of repeated samples drawn is at the researcher&#8217;s discretion. Note that bootstrap resampling is done with replacement.<\/p>\n<p>Furthermore, we can calculate the standard error of the sample mean. This is done by resampling and calculating the mean of each sample. The following formula is used to estimate the standard error.<\/p>\n<p>$$ s_{\\bar{X}}=\\sqrt{\\frac{1}{B-1}\\sum_{b=1}^{{B}}\\left({\\hat{\\theta}}_b-\\bar{\\theta}\\right)^2} $$<\/p>\n<p>Where:<\/p>\n<p>\\(s_{\\bar{X}}\\) = Estimate of the standard error of the sample mean.<\/p>\n<p>\\(B\\) = Number of resamples drawn from the original sample.<\/p>\n<p>\\({\\hat{\\theta}}_b\\) = Mean of a resample.<\/p>\n<p>\\(\\bar{\\theta}\\)= Mean across all the resample means.<\/p>\n<p>The bootstrap resampling method can also be used to estimate the confidence intervals for statistics on other population parameters, such as the median.<\/p>\n<h3>Advantages of Bootstrap Resampling<\/h3>\n<ol type=\"1\">\n<li><strong>No Reliance on Analytical Formulas<\/strong>: Bootstrap differs from traditional statistics because it doesn&#8217;t rely on an analytical formula for estimating distributions. This makes it versatile for complex estimators and especially useful when analytical formulas are unavailable.<\/li>\n<li><strong>Applicability to Complicated Estimators<\/strong>: Bootstrap is a simple yet powerful method that effectively handles complicated estimators. It can handle a wide range of statistical models, making it suitable for various applications in finance where complex estimations are common.<\/li>\n<li><strong>Increased Accuracy<\/strong>: Bootstrap can enhance accuracy by creating multiple resampled datasets and estimating population parameters on each. This helps understand estimator variability and robustness, ultimately improving result accuracy.<\/li>\n<\/ol>\n<h2>Jackknife<\/h2>\n<p>Jackknife is a resampling method in which samples are drawn by omitting one observation at a time from the original data sample. This process involves drawing samples without replacement. For a sample size of \\(n\\), we need \\(n\\) repeated samples. This method can be used to reduce the bias of an estimator or to estimate the standard error and the confidence interval of an estimator.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Assume that you are studying the median height of 100 students in a university. You draw a sample of 1000 students and obtain 1000 median heights. The mean across all resample means is 5.8. The sum of squares of the differences between each sample mean, and the mean across all resample means \\(\\sum_{b=1}^{{B}}\\left({\\hat{\\theta}}_b-\\bar{\\theta}\\right)^2\\) is 2.3.<\/p>\n<p>The Estimate of the standard error of the sample mean is <em>closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>0.05.<\/li>\n<li>0.08.<\/li>\n<li>0.10.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is A.<\/strong><\/p>\n<p>$$ \\begin{align*}<br \/>\ns_{\\bar{X}}&amp;=\\sqrt{\\frac{1}{B-1}\\sum_{b=1}^{{B}}\\left({\\hat{\\theta}}_b-\\bar{\\theta}\\right)^2} \\\\<br \/>\n&amp;=\\sqrt{\\frac{1}{1000-1}\\times 2.3}=0.04798\\approx 0.05<br \/>\n\\end{align*} $$<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Resampling refers to the act of repeatedly drawing samples from the original observed data sample for the statistical inference of population parameters. The two commonly used methods of resampling are bootstrap and jackknife. Bootstrap Using a computer, the bootstrap resampling&#8230;<\/p>\n","protected":false},"author":7,"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-45554","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>Resampling Methods Explained | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn about resampling techniques like bootstrap and jackknife, their advantages, and their role in reducing bias and estimating standard errors.\" \/>\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\/resampling-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Resampling Methods Explained | CFA Level 1 - 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