Core Algorithms

Estimate using Likelihood or Moment-Based Methods

Estimate parameters for a Hotelling

distribution with parameters p and m using maximum-likelihood or method-of-moments criteria.

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data = BlockRandom[SeedRandom[1234];

RandomVariate[HotellingTSquareDistribution[15, 20], 250]];

hist = Histogram[data, 50, "PDF", ImageSize -> 250,

ChartStyle -> Directive[Opacity[.2], Hue[.742, .7, .7]]];

momentgrid =

Grid[{{"Method of Moments"}, {"\!\(\*SubscriptBox[\(\[Mu]\), \(1\)]\

\)(data) = \!\(\*SubscriptBox[\(\[Mu]\), \(1\)]\)(dist) = " <>

ToString[

Round[Mean[

data], .01]]}, {"\!\(\*SubscriptBox[\(\[Mu]\), \

\(2\)]\)(data) = \!\(\*SubscriptBox[\(\[Mu]\), \(2\)]\)(dist) = " <>

ToString[Round[Moment[data, 2], .01]]}}, Frame -> True,

Spacings -> {3, 2}];

mldist = EstimatedDistribution[data,

HotellingTSquareDistribution[p, m]];

mmdist = EstimatedDistribution[data,

HotellingTSquareDistribution[p, m],

ParameterEstimator -> "MethodOfMoments"];

mlgrid = Grid[{{"Maximum Likelihood"}, \

{"argmax(\[ScriptL](dist,data)) = ",

MatrixForm[Round[Apply[List, mldist], .1]]}}, Frame -> True,

Spacings -> {0, 2}];

mlpdfplot =

Show[hist,

Plot[PDF[mldist, x], {x, 0, 300}, Filling -> Axis,

ImageSize -> 250, PlotStyle -> {Hue[.742, .7, .7], Thick}],

Epilog ->

Style[Text[

"p = " <> ToString[Round[mldist[[1]], .1]] <> "\nm = " <>

ToString[Round[mldist[[2]], .1]], {300, .01}],

FontFamily -> "Verdana"]];

mmpdfplot =

Show[hist,

Plot[PDF[mmdist, x], {x, 0, 300}, Filling -> Axis,

ImageSize -> 250, PlotStyle -> {Hue[.742, .7, .7], Thick}],

Epilog ->

Style[Text[

"p = " <> ToString[Round[mmdist[[1]], .1]] <> "\nm = " <>

ToString[Round[mmdist[[2]], .1]], {300, .01}],

FontFamily -> "Verdana"]];

arrow = Graphics[{Thick, {Arrowheads[Large],

Arrow[{{0, 0}, {1, 0}}, .2]}}];

Column[{Show[hist,

PlotLabel -> Style["Data", FontFamily -> "Verdana"]],

Style[Grid[{{mlgrid, arrow, mlpdfplot}, {momentgrid, arrow,

mmpdfplot}}, Spacings -> 0], FontFamily -> "Verdana"]},

Alignment -> Center]

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