2b:null 2c:["$","title",null,{"children":"Distribution of the maximum - a tale of tails | Mattias Villani | Observable"}] 31:T429,Let *X* be continuous random variable with density function ${tex`f(x)`} and distribution function ${tex`F(x)`}. The distribution of the maximum of a sample of size *n* ${tex`\qquad X_{\max}=\max(X_1,\ldots,X_n)`} from ${tex`f(x)`} has density ${tex`\qquad g(x_{\max}) = n f(x_{\max})F(x_{\max})^{n-1}`} This notebook investigates the effect of the tails of some common distributions by looking at the implied distribution of the maximum in a sample of size *n*.
The following distributions are compared: - [Normal distribution](https://observablehq.com/@mattiasvillani/normal-gaussian-distribution) - [Laplace distribution](https://observablehq.com/@mattiasvillani/laplace-distribution) (scaled to unit variance) - [Student-t distribution](https://observablehq.com/@mattiasvillani/student-t-distribution) (unscaled with variance ${tex`\nu/(\nu-2)`} for ${tex`\nu>2`}) - [Z-distribution](https://observablehq.com/@mattiasvillani/z-distribution) (${tex`\alpha=\beta=1/2`} and scaled to unit variance) All distributions have location zero and unit scale.32:T457,pltpdf = Plot.plot({ color: { legend: true, domain: ["normal", "laplace", "student-t", "z"], }, x: { label: "x", axis: true, }, y: { axis: false, label: yaxislabel, transform: (x) => {if (showlogpdf){return Math.log(x)}else{return x}} //domain: [0,1.05*jstat.normal.pdf(mu, mu, sigma)] }, title: yaxislabel, marks: [ Plot.ruleY([ylimmin]), Plot.line(pdfdata, {filter: d => d.dist == "normal" && showdist.includes("normal"), x: "x", y: "pdf", stroke : "dist", strokeWidth: 2}), Plot.line(pdfdata, {filter: d => d.dist == "student-t" && showdist.includes("student-t"), x: "x", y: "pdf", stroke : "dist", strokeWidth: 2}), Plot.line(pdfdata, {filter: d => d.dist == "laplace" && showdist.includes("laplace"), x: "x", y: "pdf", stroke : "dist", strokeWidth: 2}), Plot.line(pdfdata, {filter: d => d.dist == "z" && showdist.includes("z"), x: "x", y: "pdf", stroke : "dist", strokeWidth: 2}) ] })33:[] 2f:{"id":"6f9deb5721e706eb","slug":"distribution-of-the-maximum","title":"Distribution of the maximum - a tale of tails","thumbnail":"620d9a24a39142ef299d33efba6dc24c454cb639ee2eb04cb326c63ce0695f79","access_level":"public","create_time":"$D2023-11-05T11:04:25.197Z","update_time":"$D2025-02-11T15:12:03.680Z","view_time":null,"first_public_time":"$D2023-11-05T12:00:25.406Z","fork":{"id":"3cd309cc51f8e630","slug":"laplace-distribution","title":"Laplace distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":1,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":1128,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"3247547e8812c274","slug":"statistics","title":"Basic Statistics","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Distribution of the maximum - a tale of tails","cells":[{"id":38,"value":"# Distribution of the maximum - a tale of tails","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":706,"value":"$31","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1124,"value":"anchor = md`### Interactive exploration`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":658,"value":"viewof inputs = Inputs.form([\n Inputs.range([1, 50], {value: 10, step: 1, label: tex`\\text{sample size, }n`}),\n Inputs.range([1, 100], {value: 10, step: 1, label: tex`\\text{df student-t, }\\nu`})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":820,"value":"viewof distrinputs = Inputs.form([\n Inputs.checkbox([\"normal\", \"laplace\", \"student-t\", \"z\"], {label: \"Show distributions: \", \n value: [\"normal\", \"laplace\", \"student-t\", \"z\"]}),\n Inputs.toggle({label: \"show pdf of X in logs\", value: false})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":989,"value":"bothplots = html`\n \n \n \n

\n ${pltpdf}\n

\n ${pltmax}\n

\n \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":753,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":671,"value":"pltmax = Plot.plot({\n color: {\n legend: true,\n domain: [\"normal\", \"laplace\", \"student-t\", \"z\"]\n },\n x: {\n label: \"xₘₐₓ\",\n axis: true\n },\n y: {\n axis: false\n },\n title: \"pdf of max(X₁,...,Xₙ)\",\n marks: [\n Plot.ruleY([0]),\n Plot.line(pdfmaxdata, {filter: d => d.dist == \"normal\" && showdist.includes(\"normal\"), \n x: \"x\", y: \"maxpdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.line(pdfmaxdata, {filter: d => d.dist == \"student-t\" && showdist.includes(\"student-t\"), \n x: \"x\", y: \"maxpdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.line(pdfmaxdata, {filter: d => d.dist == \"laplace\" && showdist.includes(\"laplace\"), \n x: \"x\", y: \"maxpdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.line(pdfmaxdata, {filter: d => d.dist == \"z\" && showdist.includes(\"z\"), \n x: \"x\", y: \"maxpdf\", stroke : \"dist\", strokeWidth: 2}),\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":780,"value":"import {zpdf, zcdf, digamma, trigamma, laplacepdf, laplacecdf} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":978,"value":"showdist = distrinputs[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":980,"value":"showlogpdf = distrinputs[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":679,"value":"n = inputs[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":675,"value":"df = inputs[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":909,"value":"alpha = 1/2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":910,"value":"beta = 1/2\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":616,"value":"function invcdfsampling(invcdf, n){\n const u = d3.range(n).map(d => Math.random())\n return u\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":628,"value":"function pdfmax_studentt(x, n, df){\n return n*jstat.studentt.pdf(x, df)*jstat.studentt.cdf(x, df)**(n-1)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":633,"value":"function pdfmax_normal(x, n){\n return n*jstat.normal.pdf(x, 0, 1)*jstat.normal.cdf(x, 0, 1)**(n-1)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":637,"value":"function pdfmax_laplace(x, n){\n return n*laplacepdf(x, 0, 0.71)*laplacecdf(x, 0, 0.71)**(n-1)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":815,"value":"function pdfmax_z(x, n){\n return n*zpdf(x, alpha, beta, 0, 0.32)*zcdf(x, alpha, beta, 0, 0.32)**(n-1)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":756,"value":"pdfdata = {\n const x = d3.range(-5, 5, 0.01);\n const normal = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, 0, 1), dist: \"normal\"}));\n const student = x.map(x => ({x: x, pdf: jstat.studentt.pdf(x, df), dist: \"student-t\"}));\n const laplace = x.map(x => ({x: x, pdf: laplacepdf(x, 0, 0.71), dist: \"laplace\"}));\n const z = x.map(x => ({x: x, pdf: zpdf(x, alpha, beta, 0, 0.32), dist: \"z\"}));\n return normal.concat(student).concat(laplace).concat(z)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":673,"value":"pdfmaxdata = {\n const x = d3.range(-2, 10, 0.01);\n const normpdfmax = x.map(x => ({x: x, maxpdf: pdfmax_normal(x, n), dist: \"normal\"}));\n const studentpdfmax = x.map(x => ({x: x, maxpdf: pdfmax_studentt(x, n, df), dist: \"student-t\"}));\n const laplacepdfmax = x.map(x => ({x: x, maxpdf: pdfmax_laplace(x, n), dist: \"laplace\"}));\n const zpdfmax = x.map(x => ({x: x, maxpdf: pdfmax_z(x, n), dist: \"z\"}));\n return normpdfmax.concat(studentpdfmax).concat(laplacepdfmax).concat(zpdfmax)\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":897,"value":"z_mu = 0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":892,"value":"z_sigma = 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":887,"value":"Zmean = z_mu + z_sigma*(digamma(alpha) - digamma(beta))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":889,"value":"Zsd = Math.sqrt(z_sigma**2*(trigamma(alpha) + trigamma(beta)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":971,"value":"yaxislabel = {if (showlogpdf){return \"log pdf of X\"}else{return \"pdf of X\"}}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1020,"value":"ylimmin = {if (showlogpdf){return -50}else{return 0}}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"