2a:null 2b:["$","title",null,{"children":"Compound-Gamma distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"1ceaf1acf4ba6955","slug":"compound-gamma-distribution","title":"Compound-Gamma distribution","thumbnail":"fd1697455eafcbdfa487e848e9c136cc991ebe0f1a8ef881861cc91a5d059683","access_level":"public","create_time":"$D2025-02-16T10:49:26.542Z","update_time":"$D2025-02-16T11:39:08.176Z","view_time":null,"first_public_time":"$D2025-02-16T11:27:32.826Z","fork":{"id":"bb517b53087c21d2","slug":"gumbel-distribution","title":"Gumbel distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":1109,"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"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Compound-Gamma distribution","cells":[{"id":38,"value":"# Compound-Gamma distribution","pinned":false,"hidden":false,"mode":"md"},{"id":629,"value":"

${tex`X\\sim \\mathrm{CompoundGamma}(\\alpha,\\beta,\\kappa)\\quad \\text{ for } x>0`}","pinned":false,"hidden":false,"mode":"md"},{"id":1092,"value":"The Compound Gamma distribution (also called the **Gamma-Gamma distribution**) is obtained by the continuous mixture\n${tex.block`\\mathrm{CompoundGamma}(x\\vert\\alpha,\\beta,\\kappa) = \\int_0^\\infty \\mathrm{Gamma}(x \\vert \\kappa, \\theta) \\mathrm{Gamma}(\\theta \\vert \\alpha,\\beta)\\, \\mathrm{d}\\theta`}\nwith the [Gamma distribution](https://observablehq.com/@mattiasvillani/gamma-distribution) in the rate parameterization.","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"The density function of a ${tex`X\\sim \\mathrm{CompoundGamma}(\\alpha,\\beta,\\kappa)`} random variable is\n\n${tex.block`f(x)=\\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} \\frac{\\Gamma(\\alpha+\\kappa)}{\\Gamma(\\kappa)} \\frac{x^{\\kappa-1}}{(\\beta + x)^{\\alpha+\\kappa}}`}","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof inputs = Inputs.form({\n alpha: Inputs.range([0.01, 5], {value: 3, step: 0.01, label: tex`\\alpha`}),\n beta: Inputs.range([0.01, 5], {value: 2, step: 0.01, label: tex`\\beta`}),\n kappa: Inputs.range([0.01, 5], {value: 1, step: 0.01, label: tex`\\kappa`}),\n quantile: Inputs.range(extrema, {value: 2, step: 0.01, label: \"quantile:\"})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{CompoundGamma}(\n ${(inputs.alpha).toPrecision(2)},\n ${(inputs.beta).toPrecision(2)},\n ${(inputs.kappa).toPrecision(2)}\n ) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = {\n \n return tex`\n \\begin{aligned}\n &\\mathrm{E}( X) = ${(mean).toPrecision(3)} \\\\[0.5em]\n &\\mathrm{Stdev( X)} =${(Math.sqrt(variance)).toPrecision(3)}\\\\[0.5em]\n &P( X\\leq ${inputs.quantile}) =${cdf.toPrecision(4)}\n \\end{aligned}\n `\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true,\n //domain: support(inputs.mu, inputs.sigma)\n },\n y: {\n label: \"f(x)\",\n //axis: false,\n //domain: [0,jstat.chisquare.pdf(mode, nu1, nu2)]\n },\n marks: [\n Plot.ruleY([0]),\n //Plot.ruleX([-5]),\n Plot.line(pdfdata, {x: \"x\", y: \"pdf\", stroke : \"green\", strokeWidth: 2}),\n Plot.areaY(pdfdata, {filter: d => d.x <= inputs.quantile, x: \"x\", y: \"pdf\", fill: \"green\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is some code and data details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":750,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":899,"value":"digamma = require( 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-digamma@umd/browser.js' )","pinned":false,"hidden":false,"mode":"ojs"},{"id":828,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":909,"value":"function t(x){\n return Math.exp(-(x-inputs.mu)/inputs.beta)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":903,"value":"function pdfcompoundgamma(x, alpha, beta, kappa){\n const normconst = (beta**alpha / mathjs.gamma(alpha)) * (mathjs.gamma(alpha + kappa) / mathjs.gamma(kappa)) \n return normconst * ( (x**(kappa-1)) / ( (beta+x)**(alpha + kappa) ) )\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":907,"value":"function cdfgumbel(x, mu, beta){\n return Math.exp(-Math.exp(-(x-inputs.mu)/inputs.beta))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":850,"value":"mean = inputs.kappa*(inputs.beta/(inputs.alpha-1))","pinned":false,"hidden":false,"mode":"ojs"},{"id":766,"value":"variance = inputs.beta**2 * ( (inputs.kappa**2 + inputs.kappa*(inputs.alpha - 1)) / ( (inputs.alpha - 2)*(inputs.alpha - 1)**2 ) ) ","pinned":false,"hidden":false,"mode":"ojs"},{"id":1025,"value":"extrema = [0,15]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1078,"value":"stepsize = 0.001","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const x = d3.range(extrema[0], extrema[1], stepsize); \n const pdf = x.map(x => ({x: x, pdf: pdfcompoundgamma(x, inputs.alpha, inputs.beta, inputs.kappa)}));\n return pdf\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":19,"value":"cdf = d3.sum(pdfdata.filter(d => d.x <= inputs.quantile).map(d => d.pdf*stepsize))","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2f:"$2e:body"