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Graphical techniques that permit inferences \ regarding the distributional properties of the data are presented and \ illustrated. Procedures for compositional data in the ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["L", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox["3", FormatType->StandardForm, FontFamily->"Times", FontSize->8, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" space are examined and expanded to the ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["L", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox["1", FormatType->StandardForm, FontFamily->"Times", FontSize->8, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["constrained space for two variables and the ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["L", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox["2", FormatType->StandardForm, FontFamily->"Times", FontSize->8, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" space for three variables. Several new conventions are proposed \ that attempt to circumvent the traditional conditional approach. ", FormatType->StandardForm, FontFamily->"Times"] }], "Abstract", CellMargins->{{Inherited, 27}, {Inherited, Inherited}}, TextJustification->1, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Probability Plots-Overview", "SectionFirst", FormatType->StandardForm, FontFamily->"Times", FontSize->16], StyleBox["\n\t", FormatType->StandardForm, FontFamily->"Times"], StyleBox["Probability plots are graphical expressions used in examining \ data structures. Plots provide insights into the suitability of a particular \ probability density function (pdf) in describing the stochastic behavior of \ the data and estimates for the unknown parameters of the pdf. Although \ generally very powerful, the inferences drawn from probability plots are \ subjective. \n\tThe underlying principle behind probability plots is simple \ and consistent. The order statistics, with Y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" denoting the ith order statistic such that Y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[1", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["]", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times", FontSize->12], Cell[BoxData[ \(TraditionalForm\` \[LessEqual] \)], FontSize->12], StyleBox[" Y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[2]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times", FontSize->12], Cell[BoxData[ \(TraditionalForm\` \[LessEqual] \)], FontSize->12], StyleBox[" ... 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", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["\n\t", FormatType->StandardForm, FontFamily->"Times"], StyleBox["The expected value of the ith order statistic is\n \ E(Y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") = ", FormatType->StandardForm, FontFamily->"Times", FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"{", StyleBox[\(\(n!\)/\([\(\((i - 1)\)!\) \(\((n - i)\)!\)]\)\), FontFamily->"Times New Roman"], StyleBox["}", FontFamily->"Times New Roman"]}], SuperscriptBox[ SubscriptBox[ FormBox[ StyleBox["\[Integral]", FontSize->18], "TraditionalForm"], "0"], "1"]}], TraditionalForm]], FontSize->12], StyleBox["Y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["[F(y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[")]", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["(i-1)", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["[1-F(y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[")]", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["(n-i)", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["dF(y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["]", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") \nwhere f(y) denotes the pdf being considered, F(y) the \ associated cumulative distribution function (cdf) and n the size of the \ dataset under investigation. Because numerical solutions for this equation \ can be difficult, the approximation E(Y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") = F", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["-1", FormatType->StandardForm, FontFamily->"Times", FontSize->10, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["[(i-c)/(n-2c+1)], where F", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["-1", FormatType->StandardForm, FontFamily->"Times", FontSize->10, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" denotes the inverse cdf and c a constant (0 ", FormatType->StandardForm, FontFamily->"Times", FontSize->12], Cell[BoxData[ \(TraditionalForm\` \[LessEqual] \)], FontSize->12], StyleBox[" c ", FormatType->StandardForm, FontFamily->"Times", FontSize->12], Cell[BoxData[ \(TraditionalForm\` \[LessEqual] \)], FontSize->12], StyleBox[" 1) is frequently used. Setting c=0.5 (for discussion see \ Kimball (1960)) results in \n \ E(Y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") = F", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["-1", FormatType->StandardForm, FontFamily->"Times", FontSize->10, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["[ (i-0.5)/n] \ [1] \nand is the approximation used here. ", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["Mathematica", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], StyleBox[" will be used to evaluate E(Y", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") , create the resulting probability plot , assist in assessing \ linearity and determine parameter estimates. 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Vertical asymptotes \ are included at the 10th and 80th percentiles to facilitate determining the \ points of intersection with these asymptotes.\n\tCreating a probability plot \ for the standard uniform distribution, f(y) = 1/\[Theta], -\[Theta]/2", "Text", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\` \[LessEqual] \)], FontSize->12], " ", StyleBox["y ", "Text", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\` \[LessEqual] \)], FontSize->12], " ", StyleBox["-\[Theta]", FontSize->12], StyleBox["/2 requires changing the pdfs definition to", "Text", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", "Text", FormatType->StandardForm, FontFamily->"Times Greek", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["UniformDistribution[0, 1]", "Text", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontVariations->{"CompatibilityType"->0}], StyleBox[". In addition the PlotRange is altered to {0, 1} and asymptotes \ added at {.25, .5, .75}. If the resulting uniform probability plot is \ considered linear, then an estimate of \[Theta] is determined using the \ plot's intersection with the 25th and 75th percentiles as follows (y", "Text", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[".75", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" - y", "Text", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[".25", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[")/.5 . \n\tOther pdfs can be examined by changing the pdf \ definition in the routine. 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The Weibull distribution of \ the form f(y) =\[Alpha]\[Beta] ", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Plain"], \(\[Beta] - 1\)], TraditionalForm]], FontSize->12], StyleBox["exp[-\[Alpha]", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["y", FontSize->12, FontSlant->"Plain"], "\[Beta]"], TraditionalForm]]], StyleBox["], \[Alpha], \[Beta] >0, 0", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\` \[LessEqual] \)], FontSize->12], StyleBox["y<", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`\[Infinity]\)], "InlineFormula", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[", is an exception in that \[Alpha] and \[Beta] are considered \ scale and shape parameters. In the previous example the values of the \ parameters were set at 1 and 3.25 respectively. A linear relationship in a \ Weibull probability plot would suggest that the Weibull distribution with \ specific parameter values is appropriate in describing the stochastic nature \ of the data. However a non-linear relationship does not necessarily rule out \ the Weibull family of distributions but may be a reflection of the value(s) \ of the parameters chosen. \n\tIn the case of the Weibull distribution, \ taking the natural logarithm twice and plotting allows the distribution (with \ scale and shape parameters) to be examined analogous to those distributions \ characterized by location and scale parameters (Hahn and Shapiro (1967)). \ However, in general, probability plots can assess only those distributions \ with no (or at least a constant) shape parameter. Cheng and Spiring (1990) \ used rotation to illustrate techniques that extend the use of probability \ plots to a class of pdfs characterized by location, scale and shape \ parameters. Of particular interest were the Weibull and Tukey's-\[Lambda] \ distributions as both are characterized by a location, scale and single shape \ parameter. ", FormatType->StandardForm, FontFamily->"Times", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Text", TextJustification->1] }, Open ]], Cell[TextData[{ StyleBox["Creating & Interpreting 3-D Probability Surfaces", FormatType->StandardForm, FontFamily->"Times", FontSize->16], StyleBox["\n\t", FormatType->StandardForm, FontFamily->"Times"], StyleBox["Dynamic graphic techniques have opened new frontiers in data \ display and analysis. With a basic understanding of probability plots, \ subjective interpretation of distributional assumptions can be made for \ families of distributions that contain a shape parameter. Strong visual \ results are possible for relatively small sample sizes. In the examples that \ follow, sample sizes of 25 provide good insights into the distributional \ properties of the observed data. \n\tLet Y denote a random variable with pdf \ f(y; \[Mu], \[Sigma], \[Lambda]) and cdf F(y; \[Mu], \[Sigma], \[Lambda]) \ where \[Mu], \[Sigma] and \[Lambda] denote the location, scale and shape \ parameters of the distribution respectively. Cheng and Spiring (1990) \ defined the X-axis as E(Y", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["; \[Lambda]), scaled the Z-axis arithmetically assigning it to \ the order statistics Y", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" and let the Y axis denote values of the shape parameter \ \[Lambda], to create a surface in 3 space. Examination of the resulting \ surface allows inferences regarding the stochastic nature of the data as well \ as estimates for location, scale and shape parameters of the associated pdf. \ \n\tThe resulting surface is essentially an infinite number of traditional \ probability plots laid side by side. These probability plots are ordered by \ the value of the shape parameter used in calculating the E(Y", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[")'s. Slicing the surface along planes parallel to the XZ plane \ at various points along the Y axis, allows viewing of the \"linearity\" of \ the surface by considering the resultant projection on the XZ plane. The \ projection is a univariate probability plot of the data for a particular \ value of the shape parameter. The goal is to slice the surface such that the \ most linear projection on the XZ plane is found. \n\tRotation allows viewing \ of the created surface from several perspectives, enhancing the ability to \ determine where the surface appears most linear and the associated value of \ the shape parameter. From the most linear portion of the surface, estimates \ for the location, scale and shape parameters can be determined. The 50th \ percentile (or midpoint of the X-axis provides an estimate for the location, \ the value of the Y-axis where the surface is most linear provides an estimate \ for the shape parameter and the slope of the surface (in the X-direction) an \ estimate of the scale. In practice the order statistics are plotted versus \ the expected value of the ordered statistics for various values of the shape \ parameter. Then examining various views of the surface allows one to \ determine the value of the shape parameter associated with the most linear \ portion of the curve. From there estimates for the location and scale \ parameters are possible. \n\tAnimation permits a series of univariate \ probability plots (for specific values of the shape parameter) to be viewed \ in a sequential fashion, highlighting changes in the probability plots \ resulting from changes in the shape parameter. This results in a quick and \ reliable method for determining the most linear portion of the surface. The \ procedure creates a series of univariate probability plots representing \ various values of the shape parameter. The observer must determine which of \ the plots (if any) is most linear. If the surface provides no linear \ results, then one concludes that the data do not arise from the family of \ distributions considered. ", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Subsection", CellMargins->{{23, 0}, {Inherited, Inherited}}, TextJustification->1], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example", "Subsection", FormatType->StandardForm, FontFamily->"Times", FontSize->18], StyleBox["\n\t", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\n\tLetting y denote 25 simulated normal distribution results, \ E(Y", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["; \[Lambda]) the expected value of the associated order \ statistics and \[Lambda] the shape parameter, a surface in three space can be \ created with the following routine. 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The transformation is also the \ percentile function for the distribution and results in the expected value of \ the order statistics being of the form E(Y", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["; \[Lambda]) = (((i-0.5)/n)", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\[Lambda]", FormatType->StandardForm, FontFamily->"Times", FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" - (1-(i-0.5)/n)", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\[Lambda]", FormatType->StandardForm, FontFamily->"Times", FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[")/\[Lambda]. ", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["While rotation allows different views of this surface, \ determining the most linear portion can still be difficult. Rather than \ rotating, slicing and viewing the resulting plots , t", FontSize->12], StyleBox["he following routine creates a series of probability plots \ enhanced with the regression line and ", FormatType->StandardForm, FontFamily->"Times", FontSize->12], StyleBox["R", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["2", "Text", FormatType->StandardForm, FontFamily->"Times", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[", as well as the associated value of \[Lambda], that can be \ viewed using animation.", FormatType->StandardForm, FontFamily->"Times", FontSize->12] }], "Text", CellMargins->{{Inherited, 0}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1], Cell[BoxData[{ StyleBox[\(<< Statistics`ContinuousDistributions`\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(<< Statistics`LinearRegression`\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(y = { .1, .2025, .3045, .4124, .523, .6433, .7723, \ .9154, 1.08, 1.282, 1.555, 2.054, \n\t\(-2.054\), \(-1.555\), \(-1.282\), \(-1.08\), \(- .9154\ \), \(- .7723\), \(- .6433\), \(- .523\), \(- .4124\), \(- .3045\), \n\t\(- \ .2025\), \(- .1\), 0}; n = Length[y]; s = Min[y]; t = Max[y]; d = 4 \((t - s)\)/n;\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(l\ := \ Text[StyleForm[\ lambda\ \ "\<= \[Lambda]\>"], {\(-3\), t}];\), FontFamily->"Times New Roman"], "\n", StyleBox[\(pdfs[ j_] := \((\((\((\((j - .5)\)/n)\)^ lambda - \((1 - \((\((j - .5)\)/n)\))\)^lambda)\)/ lambda)\)\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(\(EYpairs[y_, \ pdfs_]\ := \ With[{n = Length[y]}, \n\t Transpose[{Map[ pdfs[#] &, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \n\t11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}], Sort[y]}]]\)\(\t\)\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(linePlot\ := \ \n\t Plot[Fit[EYpairs[y, \ pdfs], \ {1, x}, \ x], \n\t{x, \(-4.0\), 4.0}, DisplayFunction\ -> \ Identity];\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(r2\ := \ Text[Regress[EYpairs[y, \ pdfs], \ {1, x}, \ x, \n RegressionReport\ -> \ {RSquared}], {1.5, \(-2\)}]\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(probabilityPlot[yList_, \ pdfList_\ ]\ := \n\t ListPlot[ EYpairs[y, \ pdfs], \ \n\t\tFrameLabel\ -> \ {"\", "\"}, RotateLabel -> \ False, \n\t\tPlotRange -> {{\(-4\), 4}, {s - d, t + d}}, Frame\ -> \ True, \n\t\tDisplayFunction\ -> \ Identity];\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(Do[\(Show[probabilityPlot[yList, \ pdfList\ ], linePlot, \n Graphics[r2], Graphics[l], \n DisplayFunction\ -> \ $DisplayFunction, \n Prolog\ -> \ AbsolutePointSize[4]];\), {lambda, 0.05, 1.00, 0.05}]\), FormatType->StandardForm, FontFamily->"Times"]}], "InputOnly", FontSize->10], Cell[TextData[{ StyleBox["\t", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->10], StyleBox["The resulting series of plots represent the projections \ associated with slices of the surface taken at \[Lambda] =0.05(.05)1.0. \ Using the animation keys to sequentially examine the plots, it quickly \ becomes apparent that the most linear probability plot occurs at \[Lambda] = \ 0.15. The R", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12], StyleBox["2", FormatType->StandardForm, FontFamily->"Times", FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" value supports the visual assessment reaching its maximum of \ 0.999994 at \[Lambda] =0.15. The asymptote at E(Y", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[")=0 suggests an estimated mean of 0, while the slope of the plot \ suggests an estimated standard deviation of (0.8 - (-0.8))/2 = 0.8. This \ example highlights the relationship that exists between the normal and the \ symmetric lambda families. The symmetric lambda distribution with \ \[Lambda]=0.14 is used as an approximation to the normal distribution.\n\tThe \ pdf of the standard Weibull distribution is of the form f(y) = \[Lambda] y", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12], StyleBox["\[Lambda]-1", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12], Cell[BoxData[ \(TraditionalForm\`e\^\(-y\^\[Lambda]\)\)]], StyleBox[", ", FormatType->StandardForm, FontFamily->"Symbol", FontSize->12, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" 0StandardForm, FontFamily->"Times New Roman", FontSize->12], StyleBox["\[Infinity]", FontSize->12], StyleBox[" , and the E(Y", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12], StyleBox["[i]", FormatType->StandardForm, FontFamily->"Times", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") are approximated by ", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", RowBox[{"ln", "[", FormBox[\(\(2 n + 1 - 2 i\)\/\(2 n\)\), "TraditionalForm"], "]"}]}], ")"}], \(1\/\[Lambda]\)], TraditionalForm]]], StyleBox[". ", FontSize->12], StyleBox["The subsequent routine creates a series of univariate probability \ plots that permits examination of the Weibull family of distributions for \ values of the shape parameter \[Lambda] =1(.25)5. Again the goal is to find \ the most linear portion of the surface or most linear slice of the surface \ for the values of the shape parameter considered. In those cases where the \ \"most\" linear probability plot is deemed non-linear then either the Weibull \ family is inappropriate and/or the value of the shape parameter has not been \ included.\n\tA Weibull distribution with shape parameter of approximately \ 3.25 is often cited as a reasonable approximation to the normal distribution. \ Again using the animation keys to sequentially examine the plots, it quickly \ becomes apparent that the most linear plot of the series visually appears to \ occur at \[Lambda]=3.25 or \[Lambda]=3.50 while the R", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12], StyleBox["2", FormatType->StandardForm, FontFamily->"Times", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" value suggests that the most linear plot occurs at \ \[Lambda]=3.50.", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12] }], "Text", CellMargins->{{Inherited, -1}, {Inherited, Inherited}}, TextJustification->1], Cell[BoxData[{ StyleBox[\(<< Statistics`ContinuousDistributions`\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(<< Statistics`LinearRegression`\), FormatType->StandardForm, FontFamily->"Times"], "\n", StyleBox[\(y = { .1, .2025, .3045, .4124, .523, .6433, .7723, \ .9154, 1.08, 1.282, 1.555, 2.054, \n\t\(-2.054\), \(-1.555\), \(-1.282\), \(-1.08\), \(- \ .9154\), \(- .7723\), \(- .6433\), \(- .523\), \(- .4124\), \(- .3045\), \(- \ .2025\), \(- .1\), 0};\), FormatType->StandardForm, FontFamily->"Times"], "\[IndentingNewLine]", StyleBox[\(n = Length[y]; 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As the \ compositional make-up varies (i.e., as we observe different values of the \ triple {", "Text", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["X", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain"], StyleBox["1", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", ", "Text", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["X", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain"], StyleBox["2", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", ", "Text", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["X", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain"], StyleBox["3", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["}), the ", "Text", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["L", TextJustification->1, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Italic"], StyleBox["1", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->10, FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["a-X3", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->9, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" line will vary as will the location (i.e., {", "Text", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["X", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain"], StyleBox["1", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", ", "Text", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["X", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain"], StyleBox["2", "Text", FormatType->StandardForm, FontFamily->"Times New Roman", FontSize->9, FontWeight->"Plain", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["}) on the line. 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