File:Mean estimator consistency.gif

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Mean_estimator_consistency.gif (800 × 325 pixels, file size: 4.53 MB, MIME type: image/gif, looped, 250 frames, 25 s)

Captions

Captions

More data doesn't always lead to a better estimate.

Summary

[edit]
Description
English: If you sample from a Cauchy distribution, the probability to get a value very far on the tails is so high that more data won't ever give you a better estimate of its mean.
Date
Source https://twitter.com/j_bertolotti/status/1303999888137613312
Author Jacopo Bertolotti
Permission
(Reusing this file)		 https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 12.1 code

[edit]
n = 5 10^3;
data = RandomVariate[NormalDistribution[0, 10], n];
data2 = RandomVariate[CauchyDistribution[0, 10], n];
p0 = Table[
   GraphicsGrid[{
     {Show[
       Histogram[data[[1 ;; j]], {1}, "Probability", ChartStyle -> Orange],
       Plot[PDF[NormalDistribution[0, 10], x], {x, -40, 40}, PlotRange -> All, PlotStyle -> {Purple, Thick}]
       , PlotRange -> {{-30, 30}, {0, 0.045}}, Axes -> False, Frame -> True, FrameLabel -> {"x", "P(x)"}, LabelStyle -> {Black, Bold}, PlotLabel -> "Normal distribution"
       ]
      ,
      ListPlot[Table[Mean[data[[1 ;; i]] ], {i, 9, j}], Joined -> True, PlotRange -> {{0, n}, All}, DataRange -> j, Axes -> False, Frame -> True, FrameLabel -> {"#", "\!\(\*OverscriptBox[\(\[Mu]\), \(^\)]\)"},        LabelStyle -> {Black, Bold}, PlotStyle -> Black, Epilog -> {Purple, Thick, Dashed, Line[{{0, 0}, {n, 0}}]}]
      ,
      ListPlot[Table[StandardDeviation[data[[1 ;; i]] ], {i, 9, j}], Joined -> True, PlotRange -> {{0, n}, All}, DataRange -> j, Axes -> False, Frame -> True, FrameLabel -> {"#", "\!\(\*OverscriptBox[\(\[Sigma]\), \(^\)]\)"}, LabelStyle -> {Black, Bold}, PlotStyle -> Black, Epilog -> {Purple, Thick, Dashed, Line[{{0, 10}, {n, 10}}]}]
      }, {
      
      Show[
       Histogram[data2[[1 ;; j]], {1}, "Probability", ChartStyle -> Orange],
       Plot[PDF[CauchyDistribution[0, 10], x], {x, -40, 40}, PlotRange -> All, PlotStyle -> {Purple, Thick}], PlotRange -> {{-40, 40}, {0, 0.035}}, Axes -> False, Frame -> True, FrameLabel -> {"x", "P(x)"},        LabelStyle -> {Black, Bold}, PlotLabel -> "Cauchy distribution"]
      ,
      ListPlot[Table[Mean[data2[[1 ;; i]] ], {i, 1, j}], Joined -> True, PlotRange -> {{0, n}, All}, DataRange -> j, Axes -> False, Frame -> True, FrameLabel -> {"#", "\!\(\*OverscriptBox[\(\[Mu]\), \(^\)]\)"}, LabelStyle -> {Black, Bold}, PlotStyle -> Black, Epilog -> {Purple, Thick, Dashed, Line[{{0, 0}, {n, 0}}]}]
      ,
      ListPlot[Table[StandardDeviation[data2[[1 ;; i]] ], {i, 2, j}], Joined -> True, PlotRange -> {{0, n}, All}, DataRange -> j, Axes -> False, Frame -> True, FrameLabel -> {"#",          "\!\(\*OverscriptBox[\(\[Sigma]\), \(^\)]\)"}, LabelStyle -> {Black, Bold}, PlotStyle -> Black]
      }}, ImageSize -> 800]
   , {j, 10, n, 20}];
ListAnimate[p0]

Licensing

[edit]
I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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Date/TimeThumbnailDimensionsUserComment
current09:26, 11 September 2020Thumbnail for version as of 09:26, 11 September 2020800 × 325 (4.53 MB)Berto (talk | contribs)Uploaded own work with UploadWizard

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