(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 76863, 1756]*) (*NotebookOutlinePosition[ 77911, 1789]*) (* CellTagsIndexPosition[ 77867, 1785]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["EconGroup`NormalityTests", "Title", TextAlignment->Center], Cell["by Luci Ellis", "Subsubtitle", TextAlignment->Center], Cell["\<\ This notebook implements various tests for univariate and \ multivariate normality.\ \>", "Text"], Cell[CellGroupData[{ Cell["Reference", "Section"], Cell[CellGroupData[{ Cell["Title", "Subsubsection"], Cell[TextData[StyleBox["EconGroup`NormalityTests", FontSlant->"Italic"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Author", "Subsubsection"], Cell["Luci Ellis", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Summary", "Subsubsection"], Cell["\<\ This notebook implements various tests for univariate and \ multivariate normality.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Copyright", "Subsubsection"], Cell["\<\ \[Copyright] Copyright 1997, Luci Ellis. Permission is granted to distribute this file for any purpose except for \ inclusion in commercial software or program collections. This copyright \ notice must remain intact..\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Notebook Version", "Subsubsection"], Cell["1.0", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Version" }], "Subsubsection"], Cell["3.0", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["History", "Subsubsection"], Cell[TextData[{ "Version 1.0 for ", StyleBox["Programming in ", FontSlant->"Italic"], StyleBox["Mathematica", FontSlant->"Italic"], ", 3rd edition." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Keywords", "Subsubsection"], Cell["normality", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Source", "Subsubsection"], Cell[TextData[{ "Maeder, R. ", StyleBox["Programming in ", FontSlant->"Italic"], StyleBox["Mathematica", FontSlant->"Italic"], ", 3rd ed. Addison-Wesley, 1996." }], "Text"], Cell[TextData[{ "Urz\[UGrave]a, C. M. \[OpenCurlyDoubleQuote]On the correct uses of \ Omnibus Normality Tests\[CloseCurlyDoubleQuote], ", StyleBox["Economics Letters", FontSlant->"Italic"], ", 1997." }], "Text"], Cell[TextData[{ "Urz\[UGrave]a, C. M. \[OpenCurlyDoubleQuote]Multivariate Normality Tests\ \[CloseCurlyDoubleQuote], ", StyleBox["Advances in Econometrics", FontSlant->"Italic"], ", 1997." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Warnings", "Subsubsection"], Cell[TextData[StyleBox["description of global effects, incompatibilities", FontSlant->"Italic"]], "Text"], Cell[TextData[{ "Note: all cells marked as \"InitializationCell\" will be written to the \ Auto-Save package. This package can then be read in programs that use it with \ ", StyleBox["Needs[\"Template`\"]", "Input"], ". Cells not intended to belong to the package should not have this \ property." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Limitation", "Subsubsection"], Cell[TextData[StyleBox[ "It is assumed that the data is in matrix or vector form, with each time \ series in its own row. Therefore, the data should be presented as, say, a 3 \ by 100 list, not a 100 by 3 list.", FontSlant->"Italic"]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Discussion", "Subsubsection"], Cell[TextData[StyleBox["description of algorithm, information for experts", FontSlant->"Italic"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Requirements", "Subsubsection"], Cell["\<\ Statistics`DescriptiveStatistics`, \ Statistics`ContinuousDistributions`\ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Interface", "Section", InitializationCell->True], Cell["\<\ This part declares the publicly visible functions, options, and \ values.\ \>", "Text", InitializationCell->True], Cell[CellGroupData[{ Cell["Set up the package context, including public imports", "Subsection", InitializationCell->True], Cell["\<\ BeginPackage[\"EconGroup`NormalityTests`\", \t\"Statistics`DescriptiveStatistics`\", \ \"Statistics`ContinuousDistributions`\"]\ \>", "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["Needs[\"Statistics`ContinuousDistributions`\"]", "Input", InitializationCell->True], Cell[BoxData[ $"EconGroup`NormalityTests`"$], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Usage messages for the exported functions and the context \ itself\ \>", "Subsection", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ $NormalityTests::usage\ = \ "\"$], "Input", InitializationCell->True], Cell[BoxData[ $"NormalityTests[m] implements various tests for both univariate and \ multivariate normality."$], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[ "normalityTestStatistic::usage = \"normalityTestStatistic[x_Vector,opts] \ tests whether sample of data is normally distributed, using LM test of \ Jarque-Bera(1980) or Augmented LM test due to Urz\[UAcute]a (1997). Options \n\ are Method\[Rule]JarqueBera and Significance\[Rule]0.1\""], "Input", InitializationCell->True], Cell[BoxData[ $"Tests whether sample of data is normally distributed, using LM test of \ Jarque-Bera(1980) or Augmented LM test due to Urz\[UAcute]a (1997)."$], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ $multinormalTestStatistic::usage\ = \ "\"$], "Input", InitializationCell->True], Cell[BoxData[ $"Tests whether matrix of data is distributed according to a multinormal \ distribution."$], "Output"] }, Open ]], Cell["\<\ Method::usage = \"Method is an option for normalityTestStatistic \ and multinormalTestStatistic indicating the type of test to do. The default \ for normalityTestStatistic is JarqueBera, with Urzua being an alternative. \ The default for multinormalTestStatistic is Automatic which gives the Omnibus \ test, with alternatives of Skewness, Kurtosis and Omnibus explicitly.\"\ \>", "Input", InitializationCell->True], Cell["\<\ Significance::usage = \"Significance is an option for \ normalityTestStatistic and multinormalTestStatistic indicating the size of \ the test. For normalityTestStatistic the available sizes are 0.2, 0.15, 0.1, \ 0.05 and 0.01, while for multinormalTestStatistic the available sizes are \ 0.9, 0.95, 0.975 and 0.99.\"\ \>", "Input", InitializationCell->True], Cell["\<\ Residuals::usage = \"Residuals is an option for \ multinormalTestStatistic which is not yet implemented. The critical values \ for multivariate normality tests of residuals are not the same as for raw \ data but instead depend on the design matrix of the regression.\"\ \>", "Input", InitializationCell->True], Cell["\<\ JarqueBera::usage = \"JarqueBera is the default type of normality \ test.\"\ \>", "Input", InitializationCell->True], Cell["\<\ Urzua::usage = \"Urzua is the alternative type of normality test \ which has better small-sample properties.\"\ \>", "Input", InitializationCell->True], Cell["\<\ Omnibus::usage = \"Omnibus is the default type of multivariate \ normality test.\"\ \>", "Input", InitializationCell->True], Cell["\<\ Skewness::usage = \"Skewness is a type of multivariate normality \ test, using a multivariate skewness measure to test for normality.\"\ \>", "Input", InitializationCell->True], Cell["\<\ Kurtosis::usage = \"Kurtosis is a type of multivariate normality \ test, using a multivariate kurtosis measure to test for normality.\"\ \>", "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["Error messages for the exported objects", "Subsection", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ $normalityTestStatistic::dims\ = \ "\"$], "Input", InitializationCell->True], Cell[BoxData[ $"List of `1` dimensions entered, one-dimensional vector expected."$], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ $normalityTestStatistic::notnum\ = \ "\"$], "Input", InitializationCell->True], Cell[BoxData[ $"Input data elements must be numerical quantities."$], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ $multinormalTestStatistic::notnum\ = \ "\"$], "Input", InitializationCell->True], Cell[BoxData[ $"Input data elements must be numerical quantities."$], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ $multinormalTestStatistic::nocrit\ = \ "\"$], "Input",\ InitializationCell->True], Cell[BoxData[ $"Data must be a matrix with five or fewer rows. You may need to \ transpose the matrix prior to calling multinormalTestStatistic."$], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Implementation", "Section", InitializationCell->True], Cell["\<\ This part contains the actual definitions and any auxiliary \ functions that should not be visible outside.\ \>", "Text"], Cell[CellGroupData[{ Cell["Begin the private context (implementation part)", "Subsection", InitializationCell->True], Cell[CellGroupData[{ Cell["Begin[\"`Private`\"]", "Input", InitializationCell->True], Cell[BoxData[ $"EconGroup`NormalityTests`Private`"$], "Output"] }, Open ]], Cell["Needs[\"Statistics`ContinuousDistributions`\"]", "Input", InitializationCell->True], Cell["Off[General::spell]", "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Definition of auxiliary functions and local (static) variables\ \>", "Subsection", InitializationCell->True], Cell[CellGroupData[{ Cell["ALMCritPointTable", "Subsubsection"], Cell[BoxData[ $\(ALMCritPointTable\ = \ \n{{"\< \>", 10, 20, 35, 50, 75, 100, 150, 200, 300, 800, \[Infinity]}, \n\t\t{0.20, \ 2.14, 2.25, 2.35, 2.42, 2.62, 2.69, 2.85, 2.88, 2.92, 3.07, 3.22}, \n \t\t{0.15, 2.77, \ 2.81, 2.84, 2.95, \ 3.20, 3.16, 3.38, \ 3.44, \ 3.50, \ 3.65, \ 3.79\ }, \n \t\t{0.1, \ 4.23, \ 4.01, \ 3.81, \ \ 3.89\ , 4.17, \ 4.05, 4.26, 4.31, \ 4.32, 4.47, \ 4.61}, \n \t\t{0.05, \ 7.7, \ 7.15, \ 6.38, \ 6.43, \ 6.45, \ 6.04, \ 6.34, \ 6.22, \ 6.15, \ 6., \ 5.99}, \n \t\t{0.01, 17.69, \ 19.59, \ 16.77, \ 16.33, \ 16.32, \ 13.27, \ 13.34, \ 12.62, \ 12.1, \ 10.68, \ 9.21}}; $\)], "Input", InitializationCell->True, FontSize->10] }, Open ]], Cell[CellGroupData[{ Cell["Multivariate ALMCritPointTable(s)", "Subsubsection"], Cell[BoxData[ \(\(OmnibusALMCritPointTable = \n \t{{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 4.12, 3.94, 3.9, 4.06, 4.21, 4.39, 4.61}, \n \t\t{0.95, 7.79, 6.84, 6.42, 6.13, 6.07, 5.9, 5.99}, \n \t\t{0.975, 12.32, 11.25, 9.33, 9.02, 8.5, 7.64, 7.38}, \n \t\t{0.99, 18.61, 18.59, 15.56, 14.44, 12.55, 10.25, 9.21}}, \n \t\t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 8.65, 8.39, 7.83, 7.98, 7.88, 7.85, 7.78}, \n \t\t{0.95, 13.36, 12.63, 11.57, 11.2, 10.94, 10., 9.49}, \n \t\t{0.975, 18.82, 18.66, 16.96, 15.96, 14.34, 12.56, 11.14}, \n \t\t{0.99, 24.99, 27.68, 26.17, 23.03, 21.42, 16.35, 13.28}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 12.96, 11.94, 11.35, 11.12, 11.01, 10.83, 10.65}, \n \t\t{0.95, 18.39, 17.35, 16.62, 15.24, 14.37, 13.24, 12.59}, \n \t\t{0.975, 23.94, 24.03, 22.8, 21.47, 18.53, 15.97, 14.45}, \n \t\t{0.99, 32.33, 33.93, 33.48, 30.57, 24.14, 20.51, 16.81}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 16.27, 15.7, 14.81, 14.69, 14., 13.56, 13.36}, \n \t\t{0.95, 22.32, 22.35, 20.43, 19.16, 18.21, 16.3, 15.51}, \n \t\t{0.975, 28.83, 30.18, 27.49, 24.63, 23.03, 19.17, 17.54}, \n \t\t{0.99, 37.07, 40.82, 39.5, 34.01, 30.97, 23.46, 20.09}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 19.98, 18.67, 18.06, 17.60, 17., 16.21, 15.99}, \n \t\t{0.95, 26.99, 25.71, 23.95, 22.91, 21.25, 19.29, 18.31}, \n \t\t{0.975, 33.45, 34.01, 31.24, 28.68, 26.75, 22.24, 20.48}, \n \t\t{0.99, 42.49, 48.24, 44.01, 38.34, 33.21, 26.42, 23.21}}}; \)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(SkewnessALMCritPointTable = \n \t{{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 2.70, 2.67, 2.67, 2.61, 2.63, 2.67, 2.71}, \n \t\t{0.95, 4.14, 3.96, 3.85, 3.89, 3.83, 3.85, 3.84}, \n \t\t{0.975, 5.68, 5.45, 5.18, 5.19, 5.07, 4.88, 5.02}, \n \t\t{0.99, 7.59, 7.66, 7.1, 7.46, 6.82, 6.35, 6.64}}, \n \t\t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 4.77, 4.79, 4.64, 4.59, 4.65, 4.63, 4.61}, \n \t\t{0.95, 6.42, 6.34, 6.34, 6.15, 6.1, 6.06, 5.99}, \n \t\t{0.975, 8.04, 8.21, 7.93, 7.78, 7.77, 7.44, 7.38}, \n \t\t{0.99, 9.94, 10.64, 10.41, 10.04, 10.15, 9.21, 9.21}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 6.57, 6.49, 6.38, 6.27, 6.27, 6.14, 6.25}, \n \t\t{0.95, 8.47, 8.46, 8.33, 8.01, 7.78, 7.75, 7.82}, \n \t\t{0.975, 10.28, 10.27, 10.42, 9.77, 9.5, 9.26, 9.39}, \n \t\t{0.99, 12.69, 13.14, 13.23, 12.28, 11.82, 11.11, 11.35}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 8.17, 8.07, 8.04, 7.95, 7.93, 7.81, 7.78}, \n \t\t{0.95, 10.36, 10.38, 10.18, 9.86, 9.7, 9.57, 9.49}, \n \t\t{0.975, 12.26, 12.68, 12.19, 11.77, 11.42, 11.09, 11.14}, \n \t\t{0.99, 15.09, 15.58, 15.28, 14.27, 13.98, 13.37, 13.28}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 9.83, 9.57, 9.45, 9.44, 9.31, 9.27}, \n \t\t{0.95, 12.16, 11.9, 11.83, 11.48, 11.31, 11.1, 11.07}, \n \t\t{0.975, 14.1, 14.42, 14.03, 13.64, 13.18, 12.93, 12.83}, \n \t\t{0.99, 17.42, 17.88, 16.99, 16.7, 15.88, 15.5, 15.09}}}; \)\)], "Input", InitializationCell->True], Cell[BoxData[ \(\(KurtosisALMCritPointTable = \n \t{{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 2.07, 1.9, 1.98, 2.12, 2.31, 2.47, 2.71}, \n \t\t{0.95, 4.06, 3.65, 3.26, 3.41, 3.5, 3.62, 3.84}, \n \t\t{0.975, 6.91, 6.53, 5.51, 5.39, 4.97, 5.02}, \n \t\t{0.99, 11.29, 12.03, 10., 9.02, 8.51, 7.23, 6.64}}, \n \t\t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 4.3, 4.22, 4.07, 4.19, 4.42, 4.53, 4.61}, \n \t\t{0.95, 7.34, 7.23, 6.8, 6.96, 6.4, 6.03, 5.99}, \n \t\t{0.975, 11.11, 11.3 .3, 10.82, 10.3, 9.38, 8.02, 7.38}, \n \t\t{0.99, 19.93, 22.28, 22.34, 21.59, 17.67, 14.75, 11.35}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 6.88, 6.24, 6.01, 6.07, 6.13, 6.34, 6.25}, \n \t\t{0.95, 10.33, 10.18, 9.63, 9.2, 8.53, 8.29, 7.82}, \n \t\t{0.975, 14.28, 14.96, 14.2, 13.69, 11.8, 10.55, 9.35}, \n \t\t{0.99, 19.93, 22.28, 22.34, 21.59, 17.67, 14.75, 11.35}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 8.55, 8.67, 8.03, 8.09, 7.93, 7.81, 7.78}, \n \t\t{0.95, 12.43, 13.04, 12.2, 11.68, 10.93, 9.99, 9.49}, \n \t\t{0.975, 17.06, 19.15, 17.29, 15.92, 15.29, 12.46, 11.14}, \n \t\t{0.99, 23.2, 27.77, 26.65, 24.45, 21.53, 16.03, 13.28}}, \n \t{{"\< \>", 10, 20, 50, 100, 200, 800, \[Infinity]}, \n \t\t{0.9, 10.69, 10.16, 9.89, 9.77, 9.53, 9.25, 9.24}, \n \t\t{0.95, 15.19, 15.09, 14.37, 13.66, 12.97, 11.7, 11.07}, \n \t\t{0.975, 20., 21.41, 20.12, 18.15, 16.79, 14.25, 12.83}, \n \t\t{0.99, 26.16, 31.82, 29.09, 26.33, 22.28, 17.82, 15.09}}}; \)\)], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["ALMCritPoint functions", "Subsubsection"], Cell["\<\ ALMCritPoint[n_, sig_] := ALMCritPointTable[[Flatten[Position[ALMCritPointTable, sig]][[1]],Flatten[Position[ALMCritPointTable, n]][[ 2]] ]]\ \>", "Input", InitializationCell->True], Cell[BoxData[ $\(ALMCritPoint[n_, sig_, method_, rowdim_] := \n\t Which[method === Automatic, OmnibusALMCritPointTable\[LeftDoubleBracket]rowdim\ , \ \(Flatten[Position[OmnibusALMCritPointTable, sig]] $\[LeftDoubleBracket]2\[RightDoubleBracket], $Flatten[Position[OmnibusALMCritPointTable, n]] $\[LeftDoubleBracket]3\[RightDoubleBracket]\ \ \[RightDoubleBracket], \n\t\tmethod === Omnibus, \n\t\t OmnibusALMCritPointTable\[LeftDoubleBracket]rowdim, $Flatten[Position[OmnibusALMCritPointTable, sig]] $\[LeftDoubleBracket]2\[RightDoubleBracket], $Flatten[Position[OmnibusALMCritPointTable, n]] $\[LeftDoubleBracket]3\[RightDoubleBracket]\ \ \[RightDoubleBracket], \n\t\tmethod === Skewness, SkewnessALMCritPointTable\[LeftDoubleBracket]rowdim, $Flatten[Position[SkewnessALMCritPointTable, sig]] $\[LeftDoubleBracket]2\[RightDoubleBracket], $Flatten[Position[SkewnessALMCritPointTable, n]] $\[LeftDoubleBracket]3\[RightDoubleBracket]\ \ \[RightDoubleBracket], \n\t\tmethod === Kurtosis, KurtosisALMCritPointTable\[LeftDoubleBracket]rowdim, $Flatten[Position[KurtosisALMCritPointTable, sig]] $\[LeftDoubleBracket]2\[RightDoubleBracket], $Flatten[Position[KurtosisALMCritPointTable, n]] $\[LeftDoubleBracket]3\[RightDoubleBracket]\ \[RightDoubleBracket]]\n\t\)\)], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Definition of the exported functions", "Subsection", InitializationCell->True], Cell[CellGroupData[{ Cell["normalityTestStatistic[]", "Subsubsection", InitializationCell->True], Cell[CellGroupData[{ Cell[TextData[ "Options[normalityTestStatistic]={Method\[Rule]JarqueBera,Significance\[Rule]\ 0.1}"], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{"{", RowBox[{$Method \[Rule] JarqueBera$, ",", RowBox[{"Significance", "\[Rule]", StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]}]}], "}"}]], "Output"] }, Open ]], Cell[BoxData[ $normalityTestStatistic[x_List, opts___] := \n\t Module[{way, n, xbar, teststat, sig, ntest}, \n\t\t If[Length[Dimensions[x]] == 1, \n\t\t If[And\ @@\((\(NumericQ[#]&$\ /@\ x)\)\t, \n\t\t\t\t\t way\ = \ $(Method\ /. {opts})$ /. Options[normalityTestStatistic]; \n \t\t\tsig\ = \ $(Significance /. \ {opts})$\ /. Options[normalityTestStatistic]; \n\t\t\tn = N[Length[x]]; \n \t\t\tntest\ = \ $If[n > 2000, \[Infinity], Part[{10, 20, 35, 50, 75, 100, 150, 200, 300, 800}, Flatten[ Position[ Union[{10, 20, 35, 50, 75, 100, 150, 200, 300, 800}, {n}], n]] - 1]]$[$[1]$]; \n\t\t\t xbar\ = \ Mean[x]; \n\t\t\t If[way === Urzua, \n\t\t\t\t teststat = $(\(Plus@@\((\(#^3\ &$\ /@$(x - xbar)$)\)/n \)\/$(Plus@@\((\(#^2\ &$\ /@$(x - xbar)$)\)/n) \)\^$3\/2$)\)\^2\ \[Times]$\((n + 1)$ $(n + 3)$\)\/$6 \((n - 2)$\) + $(\(Plus@@\((\(#^4\ &$\ /@$(x - xbar)$)\)/n\)/ $(Plus@@\((\(#^2\ &$\ /@$(x - xbar)$)\)/n)\)^2 - 3 $ n - 1$\/$n + 1$)\)^2 \[Times]$\(\((n + 1)$\^2\) $(n + 3)$ $(n + 5)$\)\/$24\ n \((n - 2)$ $(n - 3)$\); \n \t\t\t\t\t TableForm[{{"\", "\", StringJoin["\", ToString[sig]]}, { teststat, ntest, ALMCritPoint[ntest, sig]}}], \n \t\t\t\t\t\t\t\t\t teststat = n $(1\/6\[Times]\(\(( \(Plus@@\((\(#^3\ &$\ /@$(x - xbar)$)\)/n\)/ $(Plus@@\((\(#^2\ &$\ /@$(x - xbar)$)\)/n) \)^$(3\/2)$)\)\ \)^2 + 1\/24\[Times]$( \(\ Plus@@\((\(#^4\ &$\ /@$(x - xbar)$)\)/n\)/ $(Plus@@\((\(#^2\ &$\ /@$(x - xbar)$)\)/n) \)^2 - 3)\)^2)\)]; \n\t\t\t\t TableForm[{{"\", "\", StringJoin["\", ToString[sig]]}, { teststat, n, Quantile[ Statistics`NormalDistribution`ChiSquareDistribution[2], 1 - sig]}}], \ Message[normalityTestStatistic::notnum]], Message[normalityTestStatistic::dims, Length[Dimensions[x]]]]]\)], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["multinormalTestStatistic[]", "Subsubsection", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ $Options[multinormalTestStatistic] = {Method \[Rule] Automatic, Significance \[Rule] 0.9, Residuals \[Rule] False}$], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{"{", RowBox[{$Method \[Rule] Automatic$, ",", RowBox[{"Significance", "\[Rule]", StyleBox["0.9`", StyleBoxAutoDelete->True, PrintPrecision->1]}], ",", $Residuals \[Rule] False$}], "}"}]], "Output"] }, Open ]], Cell[BoxData[ RowBox[{$multinormalTestStatistic[x_?MatrixQ, opts___]$, ":=", RowBox[{"If", "[", RowBox[{$And@@\(\((NumericQ[#1]&)$/@Flatten[x]\)\), ",", RowBox[{"If", "[", RowBox[{ $\(Dimensions[x]$\[LeftDoubleBracket]1\[RightDoubleBracket] < 6 \), ",", RowBox[{"Module", "[", RowBox[{ ${n, xbar, teststat, sig, ntest, way, \[DoubleStruckCapitalS], \[ScriptCapitalS], \[ScriptCapitalG], \[ScriptCapitalD], \[DoubleStruckY]}$, ",", RowBox[{ $way = \(Method /. {opts}$ /. Options[multinormalTestStatistic]\), ";", $sig = \(Significance /. {opts}$ /. Options[multinormalTestStatistic]\), ";", $n = N[ \(Dimensions[x]$\[LeftDoubleBracket]2 \[RightDoubleBracket]]\), ";", $dims = \(Dimensions[x]$\[LeftDoubleBracket]1 \[RightDoubleBracket]\), ";", $ntest = If[n > 2000, \[Infinity], \({10, 20, 50, 100, 200, 800} \[LeftDoubleBracket]Flatten[ Position[{10, 20, 50, 100, 200, 800} \[Union] {n}, n]] - 1 \[RightDoubleBracket]$\[LeftDoubleBracket]1 \[RightDoubleBracket]]\), ";", $xbar = \((Mean[#1]&)$/@x\), ";", $\[DoubleStruckCapitalS] = \(1\/n\(( Table[x\[LeftDoubleBracket]i\[RightDoubleBracket] - xbar\[LeftDoubleBracket]i \[RightDoubleBracket], {i, 1, dims}].Transpose[ Table[x\[LeftDoubleBracket]i \[RightDoubleBracket] - xbar\[LeftDoubleBracket]i \[RightDoubleBracket], {i, 1, dims}]]) $\)\), ";", $\[ScriptCapitalG] = Transpose[Eigenvectors[\[DoubleStruckCapitalS]]]$, ";", $\[ScriptCapitalD] = DiagonalMatrix[Eigenvalues[\[DoubleStruckCapitalS]]]$, ";", $\[ScriptCapitalS] = \[ScriptCapitalG]. \((\@#1&)$/@Inverse[\[ScriptCapitalD]].Transpose[ \[ScriptCapitalG]]\), ";", $\[DoubleStruckY] = \[ScriptCapitalS].Table[ x\[LeftDoubleBracket]i\[RightDoubleBracket] - xbar\[LeftDoubleBracket]i\[RightDoubleBracket], { i, 1, dims}]$, ";", $Which[way === Automatic, \(teststat = \(Plus@@Table[ \((Plus@@\(\((#1\^3&)$/@ \[DoubleStruckY]\[LeftDoubleBracket]i \[RightDoubleBracket]\)\/n)\)\^2, {i, 1, dims}]\ $(n + 1)$\ $(n + 3)$\)\/$6 \ \((n - 2)$\) + Plus@@Table[ $\((Plus@@\(\((#1\^4\/n&)$/@ \[DoubleStruckY]\[LeftDoubleBracket]i \[RightDoubleBracket]\) - $3\ \((n - 1)$\)\/$n + 1$)\)\^2\ $(\((n + 1)$\^2\ $(n + 3)$\ $(n + 5)$) \)\)\/$24\ \((n\ \((n - 2)$)\)\ $(n - 3)$\), {i, 1, dims}]; \), way === Omnibus, $teststat = \(Plus@@Table[ \((Plus@@ \(\((#1\^3&)$/@ \[DoubleStruckY]\[LeftDoubleBracket]i \[RightDoubleBracket]\)\/n)\)\^2, {i, 1, dims}]\ $(n + 1)$\ $(n + 3)$\)\/$6 \ \((n - 2)$\) + Plus@@Table[ $\((Plus@@\(\((#1\^4\/n&)$/@ \[DoubleStruckY]\[LeftDoubleBracket]i \[RightDoubleBracket]\) - $3\ \((n - 1)$\)\/$n + 1$)\)\^2\ $(\((n + 1)$\^2\ $(n + 3)$\ $(n + 5)$) \)\)\/$24\ \((n\ \((n - 2)$)\)\ $(n - 3)$\), {i, 1, dims}]; \), way === Skewness, $teststat = \(Plus@@ Table[\(( Plus@@ \(\((#1\^3&)$/@ \[DoubleStruckY]\[LeftDoubleBracket]i \[RightDoubleBracket]\)\/n)\)\^2, {i, 1, dims}]\ $(n + 1)$\ $(n + 3)$\)\/$6\ \((n - 2)$\); \), way === Kurtosis, $teststat = Plus@@Table[ \(\((Plus@@\(\((#1\^4\/n&)$/@ \[DoubleStruckY]\[LeftDoubleBracket]i \[RightDoubleBracket]\) - $3\ \((n - 1)$\)\/$n + 1$)\)\^2\ $(\((n + 1)$\^2\ $(n + 3)$\ $(n + 5)$) \)\)\/$24\ \((n\ \((n - 2)$)\)\ $(n - 3)$\), {i, 1, dims}]; \)]\), ";", TagBox[GridBox[{ {$ToString[way]<>" Test Stat"$, $"N for crit"$, $"Crit.Value at "<>ToString[sig]$}, {"teststat", "ntest", $ALMCritPoint[ntest, sig, way, dims]$} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]}]}], "]"}], ",", $Message[multinormalTestStatistic::"nocrit"]$}], "]"}], ",", $Message[multinormalTestStatistic::"notnum"]$}], "]"}]}]], "Text"], Cell[TextData[ "multinormalTestStatistic[(x_)?MatrixQ, opts___] := \n If[And @@ \ (NumericQ[#1] & ) /@ Flatten[x], \n If[Dimensions[x][[1]] < 6, \n \ Module[{n, xbar, teststat, sig, ntest, way, \n \[DoubleStruckCapitalS], \ \[ScriptCapitalS], \[ScriptCapitalG], \[ScriptCapitalD], \[DoubleStruckY]}, \n\ way = Method /. {opts} /. \n Options[multinormalTestStatistic]; \n\ sig = Significance /. {opts} /. \n \ Options[multinormalTestStatistic]; \n n = N[Dimensions[x][[2]]]; \n \ dims = Dimensions[x][[1]]; \n ntest = If[n > 2000, Infinity, \n \ {10, 20, 50, 100, 200, 800}[[Flatten[Position[Union[{10, 20, 50, 100, \ 200,800}, {n}], n]] - 1]][[1]]]; \n xbar = (Mean[#1] & ) /@ x; \n \ \[DoubleStruckCapitalS] = Table[x[[i]] - xbar[[i]], {i, 1, dims}] . \n \ Transpose[Table[x[[i]] - xbar[[i]], {i, 1, dims}]]/n; \n \ \[ScriptCapitalG] = Transpose[Eigenvectors[\[DoubleStruckCapitalS]]]; \n \ \[ScriptCapitalD] = DiagonalMatrix[Eigenvalues[\[DoubleStruckCapitalS]]]; \n\ \[ScriptCapitalS] = \[ScriptCapitalG] .(Sqrt[#1] & ) /@ \ Inverse[\\[ScriptCapitalD]] . \n Transpose[\\[ScriptCapitalG]]; \n \ \\[DoubleStruckY] = \n \\[ScriptCapitalS] . \n Table[x[[i]] - \ xbar[[i]], {i, 1, dims}]; \n Which[way === Automatic, \n teststat \ = \n Plus @@ Table[(Plus @@ (#1^3 & ) /@ \\[DoubleStruckY][[i]]/n\n\ )^2, {i, 1, dims}]*(n + 1)*(n + 3)/(6*(n - 2)) + \n Plus @@ \ Table[(Plus @@ \n (#1^4/n & ) /@ \\[DoubleStruckY][[i]] \\\n- \ (3*(n - 1))/(n + 1))^2*\n ((n + 1)^2*(n + 3)*(n + 5))/\n \ (24*n*(n - 2)*(n - 3)), {i, 1, dims}]; , \n way === Omnibus, \n \ teststat = Plus @@ Table[(Plus @@ \n (#1^3 & ) /@ \ \\[DoubleStruckY][[i]]/n\n)^2, {i, 1, dims}]*(n + 1)*(n + 3)/(6*(n - 2)) + \n \ Plus @@ \n Table[(Plus @@ \n (#1^4/n & ) \ /@ \\[DoubleStruckY][[i]] \\\n- (3*(n - 1))/(n + 1))^2*\n ((n + \ 1)^2*(n + 3)*(n + 5))/\n (24*n*(n - 2)*(n - 3)), {i, 1, dims}]; \ , \n way === Skewness, \n teststat = \n Plus @@ \n \ Table[(Plus @@ \n (#1^3 & ) /@ \ \\[DoubleStruckY][[i]]/n)^2\\\n, {i, 1, dims}]*(n + 1)*(n + 3)/(6*(n - 2)); , \ \n way === Kurtosis, \n teststat = \n Plus @@ \n \ Table[(Plus @@ \n (#1^4/n & ) /@ \\[DoubleStruckY][[i]] - \n \ (3*(n - 1))/(n + 1))^2*\n ((n + 1)^2*(n + 3)*(n + 5))/\ \n (24*n*(n - 2)*(n - 3)), {i, 1, dims}]; ]; \n \ TableForm[{{StringJoin[ToString[way], \n \" Test Stat\"], \"N for \ crit\", \n StringJoin[\"Crit.Value at \", ToString[sig]]}, \n \ {teststat, ntest, \n ALMCritPoint[ntest, sig, way, dims]}}]], \n \ Message[multinormalTestStatistic::nocrit]], \n \ Message[multinormalTestStatistic::notnum]]"], "Input", InitializationCell->True] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["End the private context", "Subsection", InitializationCell->True], Cell["On[General::spell]", "Input", InitializationCell->True], Cell[CellGroupData[{ Cell["End[ ]", "Input", InitializationCell->True], Cell[BoxData[ $"EconGroup`NormalityTests`Private`"$], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Epilog", "Section", InitializationCell->True], Cell["This section protects exported symbols and ends the package.", "Text"], Cell[CellGroupData[{ Cell["Protect exported symbol", "Subsection", InitializationCell->True], Cell[CellGroupData[{ Cell["Protect[ normalityTestStatistic, multinormalTestStatistic ]", "Input", InitializationCell->True], Cell[BoxData[ ${"normalityTestStatistic", "multinormalTestStatistic"}$], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["End the package context", "Subsection", InitializationCell->True], Cell["EndPackage[ ]", "Input", InitializationCell->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Examples, Tests", "Section"], Cell["This reads in the package", "Text"], Cell[BoxData[ $Needs["\"]$], "Input"], Cell["\<\ testdata1 is a vector of uniformly distributed random numbers \ between zero and one. testdata2 is a 3-by-500 matrix of iid uniformly \ distributed random numbers. testdata3 is a vector of numbers drawn from the \ Normal(0,2) distribution. testdata4 is a 4 by 200 vector of iid Normal(0,2) \ random numbers. Therefore the tests should reject normality for testdata1 and \ testdata2, and not reject for testdata3 and testdata4.\ \>", "Text"], Cell[BoxData[ $\(testdata1 = Table[Random[], {500}]; $\)], "Input"], Cell[BoxData[ $\(testdata2\ = \ Table[Random[], {3}, {500}]; $\)], "Input"], Cell[BoxData[ $testdata3\ = \ Table[Random[NormalDistribution[0, 2]], {200}]; \n\t\t testdata4\ = \ Table[Random[NormalDistribution[0, 2]], {4}, {200}]; $], "Input"], Cell[BoxData[ $\(toobigdata\ = \ \ Table[Random[NormalDistribution[0, 2]], {8}, {50}]; $\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ $normalityTestStatistic[testdata1, Method -> JarqueBera, Significance -> 0.05]$], "Input"], Cell[BoxData[ RowBox[{"normalityTestStatistic", "[", RowBox[{ \({0.443587977382335729`, 0.833078726509157974`, 0.232786839117430321`, 0.273311563916932254`, 0.784206055909494992`, 0.892348903424773887`, 0.348018408071448837`, 0.881454495473990107`, 0.696197554431249443`, 0.286618015650603741`, 0.845741563116715155`, 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