Scale mixture of three normal distributions

Source:

A total of 249 observations were generated by computer to follow the mixture distribution
1/3 N(12.5, 1) + 1/3 N(12.5, 3) + 1/3 N(12.5, 5)
where N(m ,s ) denotes a normal distribution with mean m and standard deviation s .

Analysis 1:

Fit a mixture of three normal distributions with means constrained to be equal and proportions constrained to be equal.

Remarks:

As with the mixture of exponential distributions, unimodality and overlap are a problem with scale mixtures. Knowing that the true proportions were equal, and so constraining the fitted proportions to be equal, helps in this example.

 Fitting Normal components

 Proportions and their standard errors
    .33333    .33333    .33333
     FIXED     FIXED     FIXED

 Means (EQUAL) and standard error
   12.4137   12.4137   12.4137
     .1466

 Sigmas and their standard errors
     .9787    3.0073    5.5181
     .1646     .5079     .5080

 Degrees of freedom = 25 - 1 +   0 -  0 -  4 -   0 =  20

 Chi-squared =  20.7126            (P =  .4142)

Analysis 2:

This is like Analysis 1, but the proportions are also fitted.

Remarks:

The estimates of the proportions and standard deviations are not very good; they have very large standard errors. The estimate of the common mean is, of course, excellent, since it is well defined on the histogram.

 Fitting Normal components

 Proportions and their standard errors
    .31122    .58136    .10742
    .10059    .18524    .21615

 Means (EQUAL) and standard error
   12.3871   12.3871   12.3871
     .1466

 Sigmas and their standard errors
     .9228    3.5607    7.8133
     .2512     .9424    5.6334

 Degrees of freedom = 25 - 1 +   0 -  0 -  6 -   0 =  18

 Chi-squared =  19.8656            (P =  .3404)
 * WARNING *  GOODNESS-OF-FIT TEST MAY BE INVALID;   4 EXPECTED COUNTS ARE < 1