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Statistics

 ProfileLikelihood
 plot a profile of the likelihood function

 Calling Sequence ProfileLikelihood(R, V, options, plotoptions) ProfileLikelihood['interactive'](R, V)

Parameters

 R - algebraic; a random variable or distribution V - options - (optional) equation(s) of the form option=value where option is one of bounds, deviations, ignore, params, range, weights; specify options for the ProfileLikelihood command plotoptions - options to be passed to the plots[display] command

Description

 • The ProfileLikelihood command plots a profile of the Likelihood command.
 • The first parameter R can be a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).
 • The second parameter V is a data sample (given as e.g. a Vector) used to calculate the likelihood function.
 • If the ['interactive'] option is used, then a dialog box appears that allows for customized creation of the plot.

Options

 The options argument can contain one or more of the options shown below. All unrecognized options will be passed to the plots[display] command. See plot/options for details.
 • bounds=none or range
 If this option specifies a range, then ProfileLikelihood attempts to calculate the maximum likelihood estimate within the given bounds. Otherwise the command attempts to calculate the maximum likelihood estimate over the entire real line (default).  Note that this option only applies if a range is not specified with the 'range' option.
 • deviations=realcons
 This option specifies the radius of the region plotted around the maximum likelihood estimate, given option range=deduce.  This option is 3 by default.
 • ignore=truefalse
 This option is used to specify how to handle non-numeric data. If ignore is set to true all non-numeric items in R are ignored.
 • params=list(name=realcons)
 This option specifies any additional indeterminates in the likelihood expression that should be evaluated prior to plotting.  The likelihood profile requires that there is only one indeterminate in the expression, so this provides a mechanism to specify the values of missing parameters.
 • range=deduce or range
 If this option is set to deduce (default), ProfileLikelihood attempts to calculate the maximum likelihood estimate and plot the likelihood function in an interval around it.  Otherwise, this parameter can specify a range to plot the likelihood profile on.
 • weights=rtable
 Vector of weights (one-dimensional rtable). If weights are given, the likelihood function is scaled so each data point has the given weight. Note that the weights provided must have type realcons and the results are floating-point, even if the problem is specified with exact values. Both the data array and the weights array must have the same number of elements.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$

Plot the profile of the likelihood function of a Pareto distribution.  These are the commands to create the plot from the Plotting Guide.

 > $S≔\mathrm{Sample}\left(\mathrm{BetaDistribution}\left(1.5,2.6\right),100\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}0.09562537057819763& 0.16216769333795644& 0.2696077416378246& 0.38505910334145665& 0.7079951962401287& 0.11451230348886078& 0.41244984761973497& 0.20124157802855755& 0.48805596029193454& 0.21126599122360598& 0.5505139935585064& 0.17026784090023775& 0.13238028678472033& 0.6285272973281937& 0.25585456016036096& 0.309132175503126& 0.09334007581879143& 0.423140332259607& 0.3166020697785465& 0.36473916983773513& 0.13942270100886286& 0.010148853186455784& 0.2548171589081302& 0.2825282228304029& 0.14555660226624365& 0.1359869588803003& 0.38431776325650885& 0.24590650524220078& 0.37981953502470733& 0.1973063052910849& 0.2560500372215065& 0.4199010264337645& 0.05807024133170289& 0.2654349623934386& 0.4827714021956183& 0.7601137232994798& 0.6209846161447357& 0.8411387615801288& 0.45160711865824027& 0.4289345542593396& 0.5643466612910764& 0.472435476047111& 0.47341626705236456& 0.4719924739972216& 0.6953361713276207& 0.15475419889971403& 0.9612784339939817& 0.4372931418376543& 0.14060985926033528& 0.677207295505058& 0.2753555789325906& 0.1572714874655496& 0.2661113834145099& 0.7245512707094466& 0.3923547349237615& 0.441572536333373& 0.25120360844939177& 0.6003019464584018& 0.2889369829510335& 0.5832935454219498& 0.6523839139895888& 0.8051248489015942& 0.5237194245787088& 0.22823572419609062& 0.5824447921976637& 0.28592646216715834& 0.2985898801260511& 0.031157102942080225& 0.3105277891843574& 0.65088547181667& 0.6846544800120201& 0.4593785331710323& 0.7321902470131976& 0.27203211965597274& 0.2732808059726067& 0.7566892227693085& 0.6182201403994758& 0.6036783462366859& 0.3446348716579453& 0.08642394754363068& 0.28520415589218345& 0.38584850330554754& 0.5009295291210871& 0.6999082446814171& 0.4210683635609472& 0.30619125425088617& 0.3366926203735391& 0.08804190754700535& 0.4533021766898777& 0.4339556665088381& 0.26914344649867555& 0.005277569100247713& 0.49667794806317894& 0.48942799454902564& 0.1138793807058265& 0.25112614267587424& 0.3827194147024285& 0.8454793997095723& 0.18169474425071114& 0.27651371015190934\end{array}\right]$ (1)
 > $R≔\mathrm{RandomVariable}\left(\mathrm{BetaDistribution}\left(1.5,\mathrm{\omega }\right)\right)$
 ${R}{≔}{\mathrm{_R0}}$ (2)
 > $\mathrm{ProfileLikelihood}\left(R,S,\mathrm{thickness}=3\right)$ > $\mathrm{ProfileLikelihood}\left(R,S,\mathrm{thickness}=3,\mathrm{color}="Red",\mathrm{axis}=\left[\mathrm{gridlines}=\left[\mathrm{linestyle}=\mathrm{dot}\right]\right]\right)$ >