31:[] 2f:{"title":"Split-t distribution","cells":[{"id":38,"value":"# Split-*t* distribution","pinned":false,"hidden":false,"mode":"md"},{"id":629,"value":"

${tex`X\\sim \\mathrm{Split}{\\text -}t(\\mu, \\tau, \\lambda, \\nu)`}, for ${tex`-\\infty \\mu\n\\end{cases}\n`}\n\nwhere ${tex`c = 2/\\big( (1 + \\lambda)\\tau\\sqrt{\\nu}\\mathrm{Beta}(\\nu/2,1/2) \\big)`}\n\n${tex`\\kappa_\\nu(x \\vert \\mu, \\tau) = \\left(1+\\frac{1}{\\nu} \\left( \\frac{ x-\\mu } {\\tau } \\right)^2\\right)^{-(\\nu+1)/2}`}\n\nis the kernel (unnormalized version) of the student-*t* density ${tex`t_{\\nu}(\\mu,\\tau^2)`}\n","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([-10, 10], {value: 0, step: 0.1, label: tex`\\text{location, }\\mu`}),\n Inputs.range([0, 10], {value: 1, step: 0.1, label: tex`\\text{scale, }\\tau`}),\n Inputs.range([0, 10], {value: 1.3, step: 0.01, label: tex`\\text{skew, }\\lambda`}),\n Inputs.range([1, 50], {value: 4, step: 1, label: tex`\\text{df, }\\nu`}),\n Inputs.range([-10, 10], {value: -1, step: 0.01, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{Split}{\\text -}t(${mu},${(tau).toPrecision(2)},${(lambda).toPrecision(2)},${(nu).toPrecision(2)}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = {\n \n return tex`\n \\begin{aligned}\n &E( X) = ${(mean).toPrecision(3)} \\\\[1em]\n &\\mathrm{Stdev}( X) = \n ${(Math.sqrt(variance)).toPrecision(3)}\\\\[1em]\n &\\mathrm{Skewness}( X) = ${(skewness).toPrecision(3)} \\\\[1em]\n &P( X\\leq ${quantile}) =${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: [-10,10]\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.line(pdfdata, {x: \"x\", y: \"pdf\", stroke : \"green\", strokeWidth: 2}),\n Plot.areaY(pdfdata, {filter: d => d.x <= 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":true,"hidden":false,"mode":"ojs"},{"id":1011,"value":"beta = require( 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-beta@umd/browser.js' )","pinned":true,"hidden":false,"mode":"ojs"},{"id":1031,"value":"// betainc(x,a,b) return the usual stats use of the incomplete beta function\nbetainc = require( 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-betainc@umd/browser.js' )","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":654,"value":"mu = params[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":656,"value":"tau = params[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":802,"value":"lambda = params[2]","pinned":false,"hidden":false,"mode":"ojs"},{"id":966,"value":"nu = params[3]","pinned":false,"hidden":false,"mode":"ojs"},{"id":662,"value":"quantile = params[4]","pinned":false,"hidden":false,"mode":"ojs"},{"id":973,"value":"h = (2*Math.sqrt(nu)*tau*(lambda-1)) / ( (nu-1)*beta(nu/2,1/2) )","pinned":false,"hidden":false,"mode":"ojs"},{"id":706,"value":"mean = mu + h","pinned":false,"hidden":false,"mode":"ojs"},{"id":766,"value":"variance = ( (1 + lambda**3)/(1 + lambda) ) * ( nu/(nu-2) )*tau**2 - h**2 ","pinned":false,"hidden":false,"mode":"ojs"},{"id":813,"value":"third_central_moment = 2*h**3 + 2*h*tau**2*(lambda**2 + 1)*(nu/(nu-3)) -3*h*tau**2*( (lambda**3 + 1)*nu ) / ( (lambda + 1)*(nu-2) ) ","pinned":false,"hidden":false,"mode":"ojs"},{"id":989,"value":"skewness = third_central_moment/variance**(3/2)","pinned":false,"hidden":false,"mode":"ojs"},{"id":897,"value":"gridsize = 0.001","pinned":false,"hidden":false,"mode":"ojs"},{"id":830,"value":"function splittpdf(x, mu, tau, lambda, nu){\n const c = 2/( (1 + lambda)*tau*Math.sqrt(nu)*beta(nu/2,1/2) )\n if (x <= mu){\n return c*( 1 + (1/nu)* ((x - mu)/tau)**2 ) **(-(nu+1)/2)\n }\n else{\n return c*( 1 + (1/nu)* ((x - mu)/(lambda*tau))**2 ) **(-(nu+1)/2)\n } \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":845,"value":"function splittcdf(x, mu, tau, lambda, nu){\n const a = x > mu ? lambda : 1\n const t = (nu*a**2*tau**2) / (nu*a**2*tau**2 + (x-mu)**2)\n return 1/(1+lambda) + ((a* Math.sign(x-mu))/(1+lambda)) * (1 - betainc(t,nu/2,1/2, false)/beta(nu/2,1/2) ) \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const x = d3.range(-10, 10, gridsize); \n const pdf = x.map(x => ({x: x, pdf: splittpdf(x, mu, tau, lambda, nu )}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"cdf = splittcdf(quantile, mu, tau, lambda, nu)","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"}