2a:null 2b:["$","title",null,{"children":"Fisher Z-distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"811384ce5401e838","slug":"z-distribution","title":"Fisher Z-distribution","thumbnail":"4d02575591c895965441364ea334bc18fa48c77867669ce551421f66006f6378","access_level":"public","create_time":"$D2023-02-24T17:01:35.649Z","update_time":"$D2026-02-17T03:28:01.627Z","view_time":null,"first_public_time":"$D2023-02-24T17:01:35.649Z","fork":{"id":"4594bdcee9954fb1","slug":"student-t-distribution","title":"Student-t distribution (three-parameter)","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":914,"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":2,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Fisher Z-distribution","cells":[{"id":38,"value":"# Fisher *Z*-distribution","pinned":false,"hidden":false,"mode":"md"},{"id":722,"value":"html`

${tex`X \\sim \\mathcal{Z}(\\alpha, \\beta, \\mu, \\sigma) \\text{ for }X\\in\\mathbb{R}, \\alpha>0, \\beta>0, \\mu \\in \\mathbb{R}, \\sigma >0\\\\[1em]`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":761,"value":"The **Fisher Z-distribution**, also called **[Type IV generalized logistic](https://en.wikipedia.org/wiki/Generalized_logistic_distribution)**, or **logistic-beta** distribution, is the distribution of the log odds of a ${tex`\\mathrm{Beta}(\\alpha,\\beta)`} variable. That is, \n\n${tex`Y \\sim \\mathrm{Beta}(\\alpha,\\beta) \\Rightarrow \\log\\Big(\\frac{Y}{1-Y}\\Big)\\sim \\mathcal{Z}(\\alpha, \\beta, 0, 1)`}\n\nThe family is location-scale so \n\n${tex`X \\sim \\mathcal{Z}(\\alpha, \\beta, 0, 1) \\Rightarrow \\mu + \\sigma X \\sim \\mathcal{Z}(\\alpha, \\beta, \\mu, \\sigma).`}","pinned":false,"hidden":false,"mode":"md"},{"id":406,"value":"### Properties","pinned":false,"hidden":false,"mode":"md"},{"id":408,"value":"moments = tex`\n\\begin{aligned}\n&E( X) = \\mu + \\sigma\\big(\\psi(\\alpha) - \\psi(\\beta)\\big) = ${Zmean.toPrecision(4)} \\\\[1em]\n&\\mathrm{stdev}( X) = \\sigma\\sqrt{\\big(\\psi^\\prime(\\alpha) +\\psi^\\prime(\\beta)\\big)} = ${Zsd.toPrecision(4)} \\\\[1em]\n&\\text{where } \\psi(\\cdot) \\text{ and } \\psi^\\prime(\\cdot) \\text{ are the digamma and trigamma functions}.\n\\end{aligned} \\\\\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof myinputs = Inputs.form([\n Inputs.range([0.1, 100], {value: 0.5, step: 0.1, label: tex`\\alpha`}),\n Inputs.range([0.1, 100], {value: 0.5, step: 0.1, label: tex`\\beta`}),\n Inputs.range([-10, 10], {value: 0, step: 0.1, label: tex`\\mu`}),\n Inputs.range([0.1, 10], {value: 1, step: 0.01, label: tex`\\sigma`}),\n Inputs.range([-10, 10], {value: -1.96, step: 0.01, label: \"quantile\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":450,"value":"viewof options = Inputs.form([\n Inputs.toggle({value: true, label: \"show closest normal\"}),\n Inputs.toggle({label: \"show pdf in logs\", value: false})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":757,"value":"tailprobs = \n{\n var tailprobs;\n if (shownormal){\n tailprobs = tex`\\text{Normal distribution: } P(X\\leq ${myinputs[4]})=${normcdf.toPrecision(4)} \\\\ \\text{Z-distribution: } P(X\\leq ${myinputs[4]})=${zcdfval.toPrecision(4)}`\n }\n else{\n tailprobs = tex`\\text{Z-distribution: } P(X\\leq ${myinputs[4]})=${zcdfval.toPrecision(4)}`\n }\n return tailprobs;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"plt = Plot.plot({\n color: {\n legend: shownormal\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n axis: false,\n label: yaxislabel,\n transform: (x) => {if (showlogpdf){return Math.log(x)}else{return x}}\n //domain: [0,1.05*jstat.normal.pdf(mu, mu, sigma)]\n },\n title: yaxislabel,\n marks: [\n Plot.ruleY([0]),\n Plot.line(data, {filter: d => d.dist == \"z\", x: \"x\", y: \"pdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.line(data, {filter: d => d.dist == \"normal\" && shownormal, x: \"x\", y: \"pdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.areaY(data, {filter: d => d.x <= myinputs[4] && d.dist == \"normal\" && shownormal, x: \"x\", y: \"pdf\", fill: \"steelblue\", opacity: 0.2}),\n Plot.areaY(data, {filter: d => d.x <= myinputs[4] && d.dist == \"z\", x: \"x\", y: \"pdf\", fill: \"orange\", opacity: 0.2}),\n \n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":828,"value":"viewof n = Inputs.range([1, 500], {value: 50, label: \"sample size, n\", step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":832,"value":"pltmax = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"z_max\",\n axis: true\n },\n y: {\n axis: false,\n //domain: [0,1.05*jstat.normal.pdf(mu, mu, sigma)]\n },\n title: \"pdf for the maximum of sample of size n\",\n marks: [\n Plot.ruleY([0]),\n Plot.line(datapdfmax, {filter: d => d.dist == \"z\", x: \"x\", y: \"pdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.line(datapdfmax, {filter: d => d.dist == \"normal\", x: \"x\", y: \"pdf\", stroke : \"dist\", strokeWidth: 2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\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"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":890,"value":"import {zpdf, zcdf, digamma, trigamma} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":874,"value":"yaxislabel = {if (showlogpdf){return \"log pdf\"}else{return \"pdf\"}}","pinned":true,"hidden":false,"mode":"ojs"},{"id":862,"value":"shownormal = options[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":863,"value":"showlogpdf = options[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":511,"value":"alpha = myinputs[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":531,"value":"beta = myinputs[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":533,"value":"mu = myinputs[2]","pinned":false,"hidden":false,"mode":"ojs"},{"id":535,"value":"sigma = myinputs[3]","pinned":false,"hidden":false,"mode":"ojs"},{"id":819,"value":"quantile = myinputs[4]","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"data = {\n const x = d3.range(-20, 20, 0.01);\n const normpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, Zmean, Zsd), dist: \"normal\"}));\n const zpdfval = x.map(x => ({x: x, pdf: zpdf(x, alpha, beta, mu, sigma), dist: \"z\"}));\n const data = normpdf.concat(zpdfval)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":735,"value":"logdata = {\n const x = d3.range(-20, 20, 0.01);\n const normpdf = x.map(x => ({x: x, pdf: Math.log(jstat.normal.pdf(x, Zmean, Zsd)), dist: \"normal\"}));\n const zpdfval = x.map(x => ({x: x, pdf: zpdf(x, alpha, beta, mu, sigma, true), dist: \"z\"}));\n const data = normpdf.concat(zpdfval)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"normcdf = jstat.normal.cdf(quantile, Zmean, Zsd);","pinned":false,"hidden":false,"mode":"ojs"},{"id":675,"value":"zcdfval = zcdf(quantile, alpha, beta, mu, sigma)","pinned":false,"hidden":false,"mode":"ojs"},{"id":834,"value":"datapdfmax = {\n const x = d3.range(0, 20, 0.01);\n const normpdf = x.map(x => ({x: x, pdf: normalpdfmax(x, Zmean, Zsd), dist: \"normal\"}));\n const zpdf = x.map(x => ({x: x, pdf: Zpdfmax(x, alpha, beta, mu, sigma), dist: \"z\"}));\n const data = normpdf.concat(zpdf)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":839,"value":"function normalpdfmax(x, mu, sigma){\n return jstat.normal.pdf(x, mu, sigma)*jstat.normal.cdf(x, mu, sigma)**(n-1)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":542,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":660,"value":"Zmean = mu + sigma*(digamma(alpha) - digamma(beta))","pinned":false,"hidden":false,"mode":"ojs"},{"id":670,"value":"Zsd = Math.sqrt(sigma**2*(trigamma(alpha) + trigamma(beta)))","pinned":false,"hidden":false,"mode":"ojs"},{"id":826,"value":"function Zpdfmax(x, alpha, beta, mu, sigma){\n return zpdf(x, alpha, beta, mu, sigma)*zcdf(x, alpha, beta, mu, sigma)**(n-1)\n}","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"