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2 changes: 1 addition & 1 deletion .nojekyll
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6 changes: 3 additions & 3 deletions asymptotics/indistribution.html

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6 changes: 3 additions & 3 deletions problemsets/04/ps04.html
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<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">

<meta name="dcterms.date" content="2023-10-13">
<meta name="dcterms.date" content="2024-10-10">

<title>ECON 626: Problem Set 4 – ECON 626</title>
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<div class="quarto-title-meta-heading">Published</div>
<div class="quarto-title-meta-contents">
<p class="date">October 13, 2023</p>
<p class="date">October 10, 2024</p>
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</div>

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<li><p>Show that <span class="math inline">\(2 \bar{X} = \frac{2}{n} \sum_{i=1}^n X_i\)</span> is an unbiased estimator for <span class="math inline">\(\theta\)</span>.</p></li>
<li><p>Show that <span class="math inline">\(\hat{\theta} = \frac{n+1}{n} \max_{1 \leq i \leq n} X_i\)</span> is an unbiased estimator for <span class="math inline">\(\theta\)</span> with <span class="math inline">\(Var(\hat{\theta}) &lt; Var(2 \bar{X})\)</span>.</p></li>
<li><p>Is there a Cramér Rao lower bound for the variance of <span class="math inline">\(\theta\)</span>? Why or why not?</p></li>
<li><p><strong>This is challenging.</strong> Show that <span class="math inline">\(\hat{\theta} = \frac{n+1}{n} \max_{1 \leq i \leq n} X_i\)</span> is a minimum variance unbiased estimator for <span class="math inline">\(\theta\)</span>.</p></li>
<li><p><strong>(Optional and challenging)</strong> Show that <span class="math inline">\(\hat{\theta} = \frac{n+1}{n} \max_{1 \leq i \leq n} X_i\)</span> is a minimum variance unbiased estimator for <span class="math inline">\(\theta\)</span>.</p></li>
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4 changes: 2 additions & 2 deletions search.json
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"href": "problemsets/04/ps04.html",
"title": "ECON 626: Problem Set 4",
"section": "",
"text": "\\[\n\\def\\R{{\\mathbb{R}}}\n\\def\\Er{{\\mathrm{E}}}\n\\]\n\nProblem 1\nThe exponential distribution has density (with respect to Lesbegue measure) \\[\nf(X;\\lambda) = \\frac{1}{\\lambda} e^{-X/\\lambda} 1\\{X &gt; 0\\}\n\\] for \\(\\lambda&gt;0\\). Suppose \\(X_1, ... , X_n\\) are independently exponential\\((\\lambda)\\) distributed .\n\nShow that the maximum likelihood estimator for \\(\\lambda\\) is \\(\\hat{\\lambda}^{MLE} = \\frac{1}{n} \\sum_{i=1}^n X_i\\)\nDerive the Cramér Rao lower bound for any unbiased estimator for \\(\\lambda\\). Is \\(\\hat{\\lambda}^{MLE}\\) a minimum variance unbiased estimator?\nFind the most powerful test of size \\(\\alpha\\) for testing \\(H_0: \\lambda = \\lambda_0\\) versus \\(H_1:\\lambda = \\lambda_1\\).\n\n\n\nProblem 2\nSuppose \\(X_1, ... , X_n\\) are independently uniformly distributed on \\((0,\\theta)\\).\n\nShow that \\(2 \\bar{X} = \\frac{2}{n} \\sum_{i=1}^n X_i\\) is an unbiased estimator for \\(\\theta\\).\nShow that \\(\\hat{\\theta} = \\frac{n+1}{n} \\max_{1 \\leq i \\leq n} X_i\\) is an unbiased estimator for \\(\\theta\\) with \\(Var(\\hat{\\theta}) &lt; Var(2 \\bar{X})\\).\nIs there a Cramér Rao lower bound for the variance of \\(\\theta\\)? Why or why not?\nThis is challenging. Show that \\(\\hat{\\theta} = \\frac{n+1}{n} \\max_{1 \\leq i \\leq n} X_i\\) is a minimum variance unbiased estimator for \\(\\theta\\)."
"text": "\\[\n\\def\\R{{\\mathbb{R}}}\n\\def\\Er{{\\mathrm{E}}}\n\\]\n\nProblem 1\nThe exponential distribution has density (with respect to Lesbegue measure) \\[\nf(X;\\lambda) = \\frac{1}{\\lambda} e^{-X/\\lambda} 1\\{X &gt; 0\\}\n\\] for \\(\\lambda&gt;0\\). Suppose \\(X_1, ... , X_n\\) are independently exponential\\((\\lambda)\\) distributed .\n\nShow that the maximum likelihood estimator for \\(\\lambda\\) is \\(\\hat{\\lambda}^{MLE} = \\frac{1}{n} \\sum_{i=1}^n X_i\\)\nDerive the Cramér Rao lower bound for any unbiased estimator for \\(\\lambda\\). Is \\(\\hat{\\lambda}^{MLE}\\) a minimum variance unbiased estimator?\nFind the most powerful test of size \\(\\alpha\\) for testing \\(H_0: \\lambda = \\lambda_0\\) versus \\(H_1:\\lambda = \\lambda_1\\).\n\n\n\nProblem 2\nSuppose \\(X_1, ... , X_n\\) are independently uniformly distributed on \\((0,\\theta)\\).\n\nShow that \\(2 \\bar{X} = \\frac{2}{n} \\sum_{i=1}^n X_i\\) is an unbiased estimator for \\(\\theta\\).\nShow that \\(\\hat{\\theta} = \\frac{n+1}{n} \\max_{1 \\leq i \\leq n} X_i\\) is an unbiased estimator for \\(\\theta\\) with \\(Var(\\hat{\\theta}) &lt; Var(2 \\bar{X})\\).\nIs there a Cramér Rao lower bound for the variance of \\(\\theta\\)? Why or why not?\n(Optional and challenging) Show that \\(\\hat{\\theta} = \\frac{n+1}{n} \\max_{1 \\leq i \\leq n} X_i\\) is a minimum variance unbiased estimator for \\(\\theta\\)."
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"text": "Asymptotic Theory of Least Squares\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nConvergence in Distribution\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nConvergence in Probability\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nDifference in Diffferences\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Final - Solutions\n\n\n\n\n\n\n\n\n\n\n\nDec 16, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Final - Solutions\n\n\n\n\n\n\n\n\n\n\n\nDec 13, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Midterm Review\n\n\n\n\n\n\n\n\n\n\n\nOct 24, 2022\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Midterm Solutions\n\n\n\n\n\n\n\n\n\n\n\nOct 26, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 1\n\n\n\n\n\n\n\n\n\n\n\nSep 12, 2024\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 2\n\n\n\n\n\n\n\n\n\n\n\nSep 23, 2024\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 3\n\n\n\n\n\n\n\n\n\n\n\nSep 30, 2024\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 4\n\n\n\n\n\n\n\n\n\n\n\nOct 13, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 5\n\n\n\n\n\n\n\n\n\n\n\nOct 20, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 6\n\n\n\n\n\n\n\n\n\n\n\nNov 9, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 7\n\n\n\n\n\n\n\n\n\n\n\nDec 1, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 7\n\n\n\n\n\n\n\n\n\n\n\nDec 6, 2022\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 8\n\n\n\n\n\n\n\n\n\n\n\nDec 6, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nEndogeneity\n\n\n\n\n\n\n\n\n\n\n\nNov 16, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nEstimation\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nGeneralized Method of Moments\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nIdentification\n\n\n\n\n\n\n\n\n\n\n\nSep 23, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nInstrumental Variables Estimation\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nLeast Squares as a Projection\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nMeasure\n\n\n\n\n\n\n\n\n\n\n\nSep 9, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nMidterm Solutions 2023\n\n\n\n\n\n\n\n\n\n\n\nOct 25, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nProbability\n\n\n\n\n\n\n\n\n\n\n\nSep 16, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\nNo matching items"
"text": "Asymptotic Theory of Least Squares\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nConvergence in Distribution\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nConvergence in Probability\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nDifference in Diffferences\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Final - Solutions\n\n\n\n\n\n\n\n\n\n\n\nDec 16, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Final - Solutions\n\n\n\n\n\n\n\n\n\n\n\nDec 13, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Midterm Review\n\n\n\n\n\n\n\n\n\n\n\nOct 24, 2022\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Midterm Solutions\n\n\n\n\n\n\n\n\n\n\n\nOct 26, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 1\n\n\n\n\n\n\n\n\n\n\n\nSep 12, 2024\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 2\n\n\n\n\n\n\n\n\n\n\n\nSep 23, 2024\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 3\n\n\n\n\n\n\n\n\n\n\n\nSep 30, 2024\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 4\n\n\n\n\n\n\n\n\n\n\n\nOct 10, 2024\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 5\n\n\n\n\n\n\n\n\n\n\n\nOct 20, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 6\n\n\n\n\n\n\n\n\n\n\n\nNov 9, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 7\n\n\n\n\n\n\n\n\n\n\n\nDec 1, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 7\n\n\n\n\n\n\n\n\n\n\n\nDec 6, 2022\n\n\n\n\n\n\n\n\n\n\n\n\nECON 626: Problem Set 8\n\n\n\n\n\n\n\n\n\n\n\nDec 6, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nEndogeneity\n\n\n\n\n\n\n\n\n\n\n\nNov 16, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nEstimation\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nGeneralized Method of Moments\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nIdentification\n\n\n\n\n\n\n\n\n\n\n\nSep 23, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nInstrumental Variables Estimation\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nLeast Squares as a Projection\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nMeasure\n\n\n\n\n\n\n\n\n\n\n\nSep 9, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nMidterm Solutions 2023\n\n\n\n\n\n\n\n\n\n\n\nOct 25, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nProbability\n\n\n\n\n\n\n\n\n\n\n\nSep 16, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\nNo matching items"
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Oct 13, 2023
Oct 10, 2024
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