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  • Fit a bivariate normal distribution from the data

    Dear all,

    How can I estimate the parameters of a bivariate normal distribution that best fit two variables in my dataset?

    Thank you!

  • #2
    Are you looking for something like the following?

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    .ÿ*
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    Estimationÿmethod:ÿml

    Logÿlikelihoodÿ=ÿ-1109.1796

    -------------------------------------------------------------------------------
    ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿOIM
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    --------------+----------------------------------------------------------------
    Structuralÿÿÿÿ|
    ÿÿy1ÿÿÿÿÿÿÿÿÿÿ|
    ÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿ-.722271ÿÿÿ.1255057ÿÿÿÿ-5.75ÿÿÿ0.000ÿÿÿÿ-.9682577ÿÿÿ-.4762842
    ÿÿ------------+----------------------------------------------------------------
    ÿÿy2ÿÿÿÿÿÿÿÿÿÿ|
    ÿÿÿÿÿÿÿÿ_consÿ|ÿÿÿ1.286987ÿÿÿ.1872411ÿÿÿÿÿ6.87ÿÿÿ0.000ÿÿÿÿÿ.9200014ÿÿÿÿ1.653973
    --------------+----------------------------------------------------------------
    ÿÿÿÿÿvar(e.y1)|ÿÿÿ3.937923ÿÿÿ.3522185ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ3.304711ÿÿÿÿ4.692465
    ÿÿÿÿÿvar(e.y2)|ÿÿÿ8.764804ÿÿÿ.7839479ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ7.355436ÿÿÿÿ10.44422
    --------------+----------------------------------------------------------------
    cov(e.y1,e.y2)|ÿÿÿ3.168499ÿÿÿ.4221588ÿÿÿÿÿ7.51ÿÿÿ0.000ÿÿÿÿÿ2.341083ÿÿÿÿ3.995915
    -------------------------------------------------------------------------------

    .ÿ
    .ÿ//ÿAlternatively
    .ÿsemÿ(y1@1ÿy2@1ÿ<-ÿF),ÿnocnsreportÿnodescribeÿnofootnoteÿnolog

    StructuralÿequationÿmodelÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿ=ÿ250
    Estimationÿmethod:ÿml

    Logÿlikelihoodÿ=ÿ-1109.1796

    ------------------------------------------------------------------------------
    ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿOIM
    ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿCoefficientÿÿstd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
    -------------+----------------------------------------------------------------
    Measurementÿÿ|
    ÿÿy1ÿÿÿÿÿÿÿÿÿ|
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    ÿÿÿÿÿÿÿ_consÿ|ÿÿÿ-.722271ÿÿÿ.1255057ÿÿÿÿ-5.75ÿÿÿ0.000ÿÿÿÿ-.9682577ÿÿÿ-.4762842
    ÿÿ-----------+----------------------------------------------------------------
    ÿÿy2ÿÿÿÿÿÿÿÿÿ|
    ÿÿÿÿÿÿÿÿÿÿÿFÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
    ÿÿÿÿÿÿÿ_consÿ|ÿÿÿ1.286987ÿÿÿ.1872411ÿÿÿÿÿ6.87ÿÿÿ0.000ÿÿÿÿÿ.9200014ÿÿÿÿ1.653973
    -------------+----------------------------------------------------------------
    ÿÿÿÿvar(e.y1)|ÿÿÿ.7694238ÿÿÿ.3203733ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.3402057ÿÿÿÿ1.740162
    ÿÿÿÿvar(e.y2)|ÿÿÿ5.596305ÿÿÿ.5902982ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ4.551103ÿÿÿÿ6.881547
    ÿÿÿÿÿÿÿvar(F)|ÿÿÿ3.168499ÿÿÿ.4221588ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ2.440297ÿÿÿÿ4.114002
    ------------------------------------------------------------------------------

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    ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿminÿ=ÿÿÿÿÿÿÿÿÿÿ2
    ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿavgÿ=ÿÿÿÿÿÿÿÿ2.0
    ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿmaxÿ=ÿÿÿÿÿÿÿÿÿÿ2
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    Logÿlikelihoodÿ=ÿ-1109.1796ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿÿÿÿÿÿÿ=ÿÿÿÿÿ0.0000

    ------------------------------------------------------------------------------
    ÿÿÿÿÿÿÿÿÿÿÿyÿ|ÿCoefficientÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
    -------------+----------------------------------------------------------------
    ÿÿÿÿÿÿÿÿÿvarÿ|
    ÿÿÿÿÿÿÿÿÿÿ1ÿÿ|ÿÿÿ-.722271ÿÿÿ.1255057ÿÿÿÿ-5.75ÿÿÿ0.000ÿÿÿÿ-.9682577ÿÿÿ-.4762842
    ÿÿÿÿÿÿÿÿÿÿ2ÿÿ|ÿÿÿ1.286987ÿÿÿ.1872411ÿÿÿÿÿ6.87ÿÿÿ0.000ÿÿÿÿÿ.9200014ÿÿÿÿ1.653973
    ------------------------------------------------------------------------------

    ------------------------------------------------------------------------------
    ÿÿRandom-effectsÿparametersÿÿ|ÿÿÿEstimateÿÿÿStd.ÿerr.ÿÿÿÿÿ[95%ÿconf.ÿinterval]
    -----------------------------+------------------------------------------------
    row:ÿIdentityÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
    ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿvar(_cons)ÿ|ÿÿÿ3.168501ÿÿÿ.4221594ÿÿÿÿÿÿ2.440298ÿÿÿÿ4.114005
    -----------------------------+------------------------------------------------
    Residual:ÿIndependent,ÿÿÿÿÿÿÿ|
    ÿÿÿÿbyÿvarÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
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    ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ2:ÿvar(e)ÿ|ÿÿÿ5.596306ÿÿÿ.5902985ÿÿÿÿÿÿ4.551103ÿÿÿÿ6.881549
    ------------------------------------------------------------------------------

    .ÿ
    .ÿexit

    endÿofÿdo-file


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