Re: st: Can't generate perfect theoretic standard normal.

[Jeph Herrin <junk@spandrel.net>]

10x does not help much - the variance is closer to 1 but
kurtosis about the same:


set mem 1500m
set obs 100000000
* 100,000,000 obs!
generate perfuniform = _n/(_N+1)
generate perfnorm = invnorm(perfuniform)
summarize perfnorm, detail

. summarize perfnorm, detail

                          perfnorm
-------------------------------------------------------------
      Percentiles      Smallest
 1%    -2.326348      -5.612001
 5%    -1.644854      -5.490852
10%    -1.281552      -5.418801       Obs            99999998
25%    -.6744897      -5.367129       Sum of Wgt.    99999998

50%    -3.74e-08                      Mean          -1.14e-07
                        Largest       Std. Dev.      .9999995
75%     .6744897       5.294704
90%     1.281551       5.294704       Variance        .999999
95%     1.644853       5.294704       Skewness      -3.31e-06
99%     2.326348       5.294704       Kurtosis       2.999973



D H wrote:
> Apparently, I need an infinite sample size to obtain a perfect
> theoretic standard normal distribution with zero skew, variance
> exactly 1 and kurtosis exactly 3.
>
> Consider the following code:
>
> version 8.2
> set memory 400m
> * 10,000,000 obs!:
> set obs 10000000
> generate double perfuniform = _n/(_N+1)
> generate double perfnorm = invnorm(perfuniform)
> summarize perfnorm, detail
>
> The last command yields:
>
>                          perfnorm
> -------------------------------------------------------------
>      Percentiles      Smallest
> 1%    -2.326346      -5.199338
> 5%    -1.644853      -5.068958
> 10%    -1.281551      -4.991217       Obs            10000000
> 25%    -.6744897      -4.935367       Sum of Wgt.    10000000
>
> 50%     6.96e-17                      Mean           7.79e-18
>                        Largest       Std. Dev.      .9999986
> 75%     .6744897       4.935367
> 90%     1.281551       4.991217       Variance       .9999971
> 95%     1.644853       5.068958       Skewness       3.23e-16
> 99%     2.326346       5.199338       Kurtosis       2.999924
>
> Now the skew is basically zero, but the kurtosis and variance are not
> precisely Gaussian.  For 20,000 observations, the above would give
> similar skew, kurtosis=2.987949 and variance = .9991686.  (n=2000:
> small skew, kurt=2.937, Var=.99386)
>
> I see a number of possible approaches to this issue.
>
> 1)  Include the numbers 1 and 0 in the uniform distribution.  This
> doesn't work: the invnorm function simply produces missing values for
> those observations.
>
> 2)  Take a harder look at my allegedly perfect uniform distribution.
> Is this where I went wrong?
>
> 3)  Try another statistics package??  (Somehow I doubt that this would 
> help.)
>
> 4)  We can always rescale the dispersion.  Divide the resulting
> distribution by the standard deviation.  Consider this addition to the
> 10,000,000 obs example:
>
> gen double perfnorm2=perfnorm/r(sd)
> sum perfnorm2, detail
>
> In that case, skew stays close to zero, var=1 and kurtosis is
> unaffected - still less than 3.
>
> 5)  Set up a loop and iteratively modify a couple of observations to
> improve the kurtosis.  Set the proper variance with 4).  I
> experimented with that: one problem is that this patch doesn't really
> produce a perfect theoretic Gaussian distribution, but rather a
> kludge.
>
> 6)  Ignore the problem and characterize it as an oddity.  Note though
> that this may imply that normally distributed standard random
> variables in Stata will not be perfectly standard normals *on
> average*: even their variance will be a little lower without
> adjustment.  These effects would presumably be swamped by ordinary
> sampling randomness.
>
> More generally, I wonder whether this is the proper way to generate a
> standard normal random variable in Stata:
>
> gen double perfuniform = _n/(_N+1)
> gen double perfnorm = invnorm(perfuniform)
> sum perfnorm
> scalar adjsd=r(sd)
> gen double randnorm = invnorm(uniform())/adjsd
>
> This procedure would not correct the kurtosis though.
>
> Might this effect matter in small samples?  Sure: for n=20, the
> unadjusted variance of the theoretic standard normal would be .794,
> which seems low.
>
> But recall point 2): I'm probably missing something.  For example, the
> uniform random number generator would not produce the sort of evenly
> spread-out numbers in my perfuniform variable.  So this adjustment may
> be inappropriate.  Monte Carlo work might clarify matters, as would
> additional conceptual insight.  In short, I would avoid using this
> adjustment without further analysis.
>
> 7)  Or perhaps I *am* generating perfectly normal distributions, but
> that theoretic normals only have kurtosis -> 3 as n -> infinity.  I
> don't know.
>
> Thanks for your attention.
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