File:Logistic-t-normal-further-tails.svg
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Summary
Comparison of standard logistic distribution with closest normal and Student's t distributions, by matching moments -- further tail portion.
Created using the following R code:
| Source Code |
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logistic.t.norm <span class="o">=</span> <span class="kr">function</span><span class="p">(</span>xmin<span class="p">,</span> xmax<span class="p">,</span> yplot.max<span class="p">,</span> filename<span class="p">)</span> <span class="p">{</span>
<span class="c1"># Generate npoints from xmin to xmax</span>
npoints<span class="o">=</span><span class="m">10000</span>
scalefact<span class="o">=</span>npoints<span class="o">/</span><span class="p">(</span>xmax <span class="o">-</span> xmin<span class="p">)</span>
x<span class="o">=</span><span class="p">(</span>xmin<span class="o">*</span>scalefact<span class="p">)</span><span class="o">:</span><span class="p">(</span>xmax<span class="o">*</span>scalefact<span class="p">)</span><span class="o">/</span>scalefact
ylim<span class="o">=</span><span class="kt">c</span><span class="p">(</span><span class="m">0</span><span class="p">,</span>yplot.max<span class="p">)</span> <span class="c1"># Minimum and maximum Y limits</span>
svg<span class="p">(</span>filename<span class="p">)</span>
<span class="c1"># cex=font scaling, mai=margins (reduced as much as possible)</span>
par<span class="p">(</span>cex<span class="o">=</span><span class="m">1.5</span><span class="p">,</span>mai<span class="o">=</span><span class="kt">c</span><span class="p">(</span><span class="m">0.8</span><span class="p">,</span><span class="m">0.8</span><span class="p">,</span><span class="m">0.2</span><span class="p">,</span><span class="m">0.1</span><span class="p">))</span>
<span class="c1"># Plot blue logistic curve; lwd=line width, type="l" means use a line</span>
plot<span class="p">(</span>x<span class="p">,</span>dlogis<span class="p">(</span>x<span class="p">),</span>type<span class="o">=</span><span class="s">"l"</span><span class="p">,</span>xlab<span class="o">=</span><span class="s">""</span><span class="p">,</span> ylab<span class="o">=</span><span class="s">""</span><span class="p">,</span>lwd<span class="o">=</span><span class="m">3</span><span class="p">,</span>col<span class="o">=</span><span class="s">"blue"</span><span class="p">,</span>ylim<span class="o">=</span>ylim<span class="p">)</span>
abline<span class="p">(</span>v<span class="o">=</span><span class="m">0</span><span class="p">,</span>lty<span class="o">=</span><span class="m">3</span><span class="p">)</span> <span class="c1"># Draw a dotted vertical line at 0 (lty=line type, 3=dotted)</span>
<span class="c1"># Set Student t params to match moments of logistic</span>
df <span class="o">=</span> <span class="m">9</span><span class="p">;</span> shape <span class="o">=</span> <span class="kp">sqrt</span><span class="p">((</span>df<span class="m">-2</span><span class="p">)</span><span class="o">/</span>df<span class="o">*</span><span class="kc">pi</span><span class="o">*</span><span class="kc">pi</span><span class="o">/</span><span class="m">3</span><span class="p">)</span>
<span class="c1"># Plot red Student t curve; lwd=line width, lty=line type (3=dotted)</span>
points<span class="p">(</span>x<span class="p">,</span>dt<span class="p">(</span>x<span class="o">/</span>shape<span class="p">,</span>df<span class="p">)</span><span class="o">/</span>shape<span class="p">,</span>type<span class="o">=</span><span class="s">"l"</span><span class="p">,</span>lty<span class="o">=</span><span class="m">3</span><span class="p">,</span>col<span class="o">=</span><span class="s">"red"</span><span class="p">,</span>lwd<span class="o">=</span><span class="m">3</span><span class="p">)</span>
<span class="c1"># Set normal params to match moments of logistic</span>
sd <span class="o">=</span> <span class="kp">sqrt</span><span class="p">(</span><span class="kc">pi</span><span class="o">*</span><span class="kc">pi</span><span class="o">/</span><span class="m">3</span><span class="p">)</span>
<span class="c1"># Draw std dev lines (currently only at +/- 1 std dev, alternatively at all of them)</span>
<span class="c1">#for (sdevs in 1:floor(xmax/sd)) {</span>
<span class="kr">for</span> <span class="p">(</span>sdevs <span class="kr">in</span> <span class="m">1</span><span class="o">:</span><span class="m">1</span><span class="p">)</span> <span class="p">{</span>
abline<span class="p">(</span>v<span class="o">=-</span>sd<span class="o">*</span>sdevs<span class="p">,</span>lty<span class="o">=</span><span class="m">2</span><span class="p">)</span> <span class="c1"># Draw a dashed vertical line at -1 std dev (lty 2=dashed)</span>
abline<span class="p">(</span>v<span class="o">=</span>sd<span class="o">*</span>sdevs<span class="p">,</span>lty<span class="o">=</span><span class="m">2</span><span class="p">)</span> <span class="c1"># Draw a dashed vertical line at +1 std dev (lty 2=dashed)</span>
<span class="p">}</span>
<span class="c1"># Add std dev ticks along the top</span>
axis<span class="p">(</span><span class="m">3</span><span class="p">,</span>at<span class="o">=-</span><span class="kp">floor</span><span class="p">(</span>xmax<span class="o">/</span>sd<span class="p">)</span><span class="o">:</span><span class="kp">floor</span><span class="p">(</span>xmax<span class="o">/</span>sd<span class="p">)</span><span class="o">*</span>sd<span class="p">,</span>labels<span class="o">=</span><span class="kc">FALSE</span><span class="p">)</span>
<span class="c1"># Plot green normal curve; lwd=line width, lty=line type (3=dotted)</span>
points<span class="p">(</span>x<span class="p">,</span>dnorm<span class="p">(</span>x<span class="p">,</span><span class="m">0</span><span class="p">,</span>sd<span class="p">),</span>type<span class="o">=</span><span class="s">"l"</span><span class="p">,</span>lty<span class="o">=</span><span class="m">3</span><span class="p">,</span>col<span class="o">=</span><span class="s">"green"</span><span class="p">,</span>lwd<span class="o">=</span><span class="m">3</span><span class="p">)</span>
<span class="c1"># Draw legend in bottom center</span>
legend<span class="p">(</span><span class="s">"bottom"</span><span class="p">,</span> <span class="kt">c</span><span class="p">(</span><span class="s">"logistic(0,1)"</span><span class="p">,</span> <span class="kp">sprintf</span><span class="p">(</span><span class="s">"t(%g,0,%g)"</span><span class="p">,</span>df<span class="p">,</span>shape<span class="p">),</span> <span class="kp">sprintf</span><span class="p">(</span><span class="s">"norm(0,%g)"</span><span class="p">,</span> sd<span class="p">),</span> <span class="kp">sprintf</span><span class="p">(</span><span class="s">"std dev=%g"</span><span class="p">,</span> sd<span class="p">)),</span> cex<span class="o">=</span><span class="m">0.9</span><span class="p">,</span>col <span class="o">=</span> <span class="kt">c</span><span class="p">(</span><span class="s">"blue"</span><span class="p">,</span><span class="s">"red"</span><span class="p">,</span><span class="s">"green"</span><span class="p">,</span> <span class="s">"black"</span><span class="p">),</span> lty<span class="o">=</span><span class="kt">c</span><span class="p">(</span><span class="m">1</span><span class="p">,</span><span class="m">3</span><span class="p">,</span><span class="m">3</span><span class="p">,</span><span class="m">2</span><span class="p">),</span>lwd<span class="o">=</span><span class="kt">c</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="m">3</span><span class="p">,</span><span class="m">3</span><span class="p">,</span><span class="m">1</span><span class="p">))</span>
dev.off<span class="p">()</span>
<span class="p">}</span>
logistic.t.norm<span class="p">(</span><span class="m">-18</span><span class="p">,</span><span class="m">18</span><span class="p">,</span><span class="m">0.0001</span><span class="p">,</span><span class="s">"logistic-t-normal-further-tails.svg"</span><span class="p">)</span>
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Licensing
Lua error in package.lua at line 80: module 'strict' not found.Lua error in package.lua at line 80: module 'strict' not found.
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 18:10, 14 January 2017 | | 630 × 630 (419 KB) | 127.0.0.1 (talk) | Comparison of standard logistic distribution with closest normal and Student's t distributions, by matching moments -- further tail portion. <p>Created using the following R code: </p> <div style="margin-left:0px"> <table class="navbox collapsible collapsed" style="background: transparent; text-align: left; border: 1px solid silver; margin-top: 0.2em;"> <tr><th style="background-color: #CFC; text-align:center; font-size:112%;"> Source Code </th></tr> <tr><td style="border: solid 1px silver; padding: 8px; background-color: white; font-size:112%;"> <div class="mw-highlight mw-content-ltr" dir="ltr"><pre>logistic.t.norm <span class="o">=</span> <span class="kr">function</span><span class="p">(</span>xmin<span class="p">,</span> xmax<span class="p">,</span> yplot.max<span class="p">,</span> filename<span class="p">)</span> <span class="p">{</span> <span class="c1"># Generate npoints from xmin to xmax</span> npoints<span class="o">=</span><span class="m">10000</span> scalefact<span class="o">=</span>npoints<span class="o">/</span><span class="p">(</span>xmax <span class="o">-</span> xmin<span class="p">)</span> x<span class="o">=</span><span class="p">(</span>xmin<span class="o">*</span>scalefact<span class="p">)</span><span class="o">:</span><span class="p">(</span>xmax<span class="o">*</span>scalefact<span class="p">)</span><span class="o">/</span>scalefact ylim<span class="o">=</span><span class="kt">c</span><span class="p">(</span><span class="m">0</span><span class="p">,</span>yplot.max<span class="p">)</span> <span class="c1"># Minimum and maximum Y limits</span> svg<span class="p">(</span>filename<span class="p">)</span> <span class="c1"># cex=font scaling, mai=margins (reduced as much as possible)</span> par<span class="p">(</span>cex<span class="o">=</span><span class="m">1.5</span><span class="p">,</span>mai<span class="o">=</span><span class="kt">c</span><span class="p">(</span><span class="m">0.8</span><span class="p">,</span><span class="m">0.8</span><span class="p">,</span><span class="m">0.2</span><span class="p">,</span><span class="m">0.1</span><span class="p">))</span> <span class="c1"># Plot blue logistic curve; lwd=line width, type="l" means use a line</span> plot<span class="p">(</span>x<span class="p">,</span>dlogis<span class="p">(</span>x<span class="p">),</span>type<span class="o">=</span><span class="s">"l"</span><span class="p">,</span>xlab<span class="o">=</span><span class="s">""</span><span class="p">,</span> ylab<span class="o">=</span><span class="s">""</span><span class="p">,</span>lwd<span class="o">=</span><span class="m">3</span><span class="p">,</span>col<span class="o">=</span><span class="s">"blue"</span><span class="p">,</span>ylim<span class="o">=</span>ylim<span class="p">)</span> abline<span class="p">(</span>v<span class="o">=</span><span class="m">0</span><span class="p">,</span>lty<span class="o">=</span><span class="m">3</span><span class="p">)</span> <span class="c1"># Draw a dotted vertical line at 0 (lty=line type, 3=dotted)</span> <span class="c1"># Set Student t params to match moments of logistic</span> df <span class="o">=</span> <span class="m">9</span><span class="p">;</span> shape <span class="o">=</span> <span class="kp">sqrt</span><span class="p">((</span>df<span class="m">-2</span><span class="p">)</span><span class="o">/</span>df<span class="o">*</span><span class="kc">pi</span><span class="o">*</span><span class="kc">pi</span><span class="o">/</span><span class="m">3</span><span class="p">)</span> <span class="c1"># Plot red Student t curve; lwd=line width, lty=line type (3=dotted)</span> points<span class="p">(</span>x<span class="p">,</span>dt<span class="p">(</span>x<span class="o">/</span>shape<span class="p">,</span>df<span class="p">)</span><span class="o">/</span>shape<span class="p">,</span>type<span class="o">=</span><span class="s">"l"</span><span class="p">,</span>lty<span class="o">=</span><span class="m">3</span><span class="p">,</span>col<span class="o">=</span><span class="s">"red"</span><span class="p">,</span>lwd<span class="o">=</span><span class="m">3</span><span class="p">)</span> <span class="c1"># Set normal params to match moments of logistic</span> sd <span class="o">=</span> <span class="kp">sqrt</span><span class="p">(</span><span class="kc">pi</span><span class="o">*</span><span class="kc">pi</span><span class="o">/</span><span class="m">3</span><span class="p">)</span> <span class="c1"># Draw std dev lines (currently only at +/- 1 std dev, alternatively at all of them)</span> <span class="c1">#for (sdevs in 1:floor(xmax/sd)) {</span> <span class="kr">for</span> <span class="p">(</span>sdevs <span class="kr">in</span> <span class="m">1</span><span class="o">:</span><span class="m">1</span><span class="p">)</span> <span class="p">{</span> abline<span class="p">(</span>v<span class="o">=-</span>sd<span class="o">*</span>sdevs<span class="p">,</span>lty<span class="o">=</span><span class="m">2</span><span class="p">)</span> <span class="c1"># Draw a dashed vertical line at -1 std dev (lty 2=dashed)</span> abline<span class="p">(</span>v<span class="o">=</span>sd<span class="o">*</span>sdevs<span class="p">,</span>lty<span class="o">=</span><span class="m">2</span><span class="p">)</span> <span class="c1"># Draw a dashed vertical line at +1 std dev (lty 2=dashed)</span> <span class="p">}</span> <span class="c1"># Add std dev ticks along the top</span> axis<span class="p">(</span><span class="m">3</span><span class="p">,</span>at<span class="o">=-</span><span class="kp">floor</span><span class="p">(</span>xmax<span class="o">/</span>sd<span class="p">)</span><span class="o">:</span><span class="kp">floor</span><span class="p">(</span>xmax<span class="o">/</span>sd<span class="p">)</span><span class="o">*</span>sd<span class="p">,</span>labels<span class="o">=</span><span class="kc">FALSE</span><span class="p">)</span> <span class="c1"># Plot green normal curve; lwd=line width, lty=line type (3=dotted)</span> points<span class="p">(</span>x<span class="p">,</span>dnorm<span class="p">(</span>x<span class="p">,</span><span class="m">0</span><span class="p">,</span>sd<span class="p">),</span>type<span class="o">=</span><span class="s">"l"</span><span class="p">,</span>lty<span class="o">=</span><span class="m">3</span><span class="p">,</span>col<span class="o">=</span><span class="s">"green"</span><span class="p">,</span>lwd<span class="o">=</span><span class="m">3</span><span class="p">)</span> <span class="c1"># Draw legend in bottom center</span> legend<span class="p">(</span><span class="s">"bottom"</span><span class="p">,</span> <span class="kt">c</span><span class="p">(</span><span class="s">"logistic(0,1)"</span><span class="p">,</span> <span class="kp">sprintf</span><span class="p">(</span><span class="s">"t(%g,0,%g)"</span><span class="p">,</span>df<span class="p">,</span>shape<span class="p">),</span> <span class="kp">sprintf</span><span class="p">(</span><span class="s">"norm(0,%g)"</span><span class="p">,</span> sd<span class="p">),</span> <span class="kp">sprintf</span><span class="p">(</span><span class="s">"std dev=%g"</span><span class="p">,</span> sd<span class="p">)),</span> cex<span class="o">=</span><span class="m">0.9</span><span class="p">,</span>col <span class="o">=</span> <span class="kt">c</span><span class="p">(</span><span class="s">"blue"</span><span class="p">,</span><span class="s">"red"</span><span class="p">,</span><span class="s">"green"</span><span class="p">,</span> <span class="s">"black"</span><span class="p">),</span> lty<span class="o">=</span><span class="kt">c</span><span class="p">(</span><span class="m">1</span><span class="p">,</span><span class="m">3</span><span class="p">,</span><span class="m">3</span><span class="p">,</span><span class="m">2</span><span class="p">),</span>lwd<span class="o">=</span><span class="kt">c</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="m">3</span><span class="p">,</span><span class="m">3</span><span class="p">,</span><span class="m">1</span><span class="p">))</span> dev.off<span class="p">()</span> <span class="p">}</span> logistic.t.norm<span class="p">(</span><span class="m">-18</span><span class="p">,</span><span class="m">18</span><span class="p">,</span><span class="m">0.0001</span><span class="p">,</span><span class="s">"logistic-t-normal-further-tails.svg"</span><span class="p">)</span> </pre></div> </td></tr> </table> </div> |
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