Generating a random normal variable with a fixed number of unique values

[Christian Agethen]

Dear all,
I'm using a data set with 5,000,000 observations and I'd like to generate a normally distributed random variable random which has (approximately) unique values: 2,000,000.
So far, I've managed to generate a uniformly distributed variable random with unique values: 1,835,364 by:

clear
set seed 2052018
set obs 5000000
gen double u = ((1999999)*runiform() + 1) * ((2^32-1)/2^32)
gen random = round(u)

Unfortunately, I can't seem to figure out how to obtain a normally distributed random that has the same number of unique values. Does anyone know a way to code this?

Best
Christian

[Nick Cox]

A distribution can't be normal and discrete. I don't know why you want this, but the corresponding binomial is likely to be a better starting point. For example,

. set seed 2803

. set obs 100000
number of observations (_N) was 0, now 100,000

. gen y = rbinomial(11, 0.5)

. tab y

          y |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |         43        0.04        0.04
          1 |        554        0.55        0.60
          2 |      2,705        2.71        3.30
          3 |      8,058        8.06       11.36
          4 |     16,171       16.17       27.53
          5 |     22,673       22.67       50.20
          6 |     22,598       22.60       72.80
          7 |     15,822       15.82       88.62
          8 |      8,158        8.16       96.78
          9 |      2,655        2.65       99.44
         10 |        514        0.51       99.95
         11 |         49        0.05      100.00
------------+-----------------------------------
      Total |    100,000      100.00

As always distinct is a better word than unique when it's what you mean. Unique still carries the primary sense of occurring just once. More at https://www.stata-journal.com/sjpdf....iclenum=dm0042

That said, I think you'll be hard pushed to get 2 million distinct values AND approximately normal distributions without a much larger sample size than you're playing with.

[William Lisowski]

I agree entirely with Nick, I cannot imagine the purpose of generating a continuous random variable and constraining it to a limited number of distinct values. With that said, another approach is to generate a full set of random values, and then coarsen them. The following code, run on my system, produces the coarsened variable y with 2,000,000 distinct values, and inspecting the summary statistics for x and y shows their distributions are essentially identical.

clear
set obs 5000000
set seed 2052018
generate double x = rnormal()
sort x
generate long group = floor((_n-1)*2/5)+1
bysort group (x): generate double  y = cond(_N==3,x[2],(x[1]+x[2])/2)
codebook x y, compact
summarize x y, detail

. codebook x y, compact

Variable       Obs  Unique       Mean        Min       Max  Label
------------------------------------------------------------------------------------------------
x          5000000 5000000  -.0003158  -4.724853  5.376398  
y          5000000 2000000  -.0003159  -4.719964  5.176187  
------------------------------------------------------------------------------------------------

. summarize x y, detail

                              x
-------------------------------------------------------------
      Percentiles      Smallest
 1%    -2.326908      -4.724853
 5%    -1.645423      -4.719964
10%    -1.282543      -4.695449       Obs           5,000,000
25%    -.6749557      -4.650745       Sum of Wgt.   5,000,000

50%    -.0002361                      Mean          -.0003158
                        Largest       Std. Dev.      1.000163
75%     .6742286       4.952557
90%     1.282528       4.969922       Variance       1.000326
95%     1.644754       4.975977       Skewness       .0003641
99%     2.325678       5.376398       Kurtosis       2.998477

                              y
-------------------------------------------------------------
      Percentiles      Smallest
 1%    -2.326908      -4.719964
 5%    -1.645423      -4.719964
10%    -1.282542      -4.719964       Obs           5,000,000
25%    -.6749554      -4.645181       Sum of Wgt.   5,000,000

50%    -.0002357                      Mean          -.0003159
                        Largest       Std. Dev.      1.000163
75%     .6742288       4.952557
90%     1.282527       4.952557       Variance       1.000326
95%     1.644754       5.176187       Skewness        .000362
99%     2.325678       5.176187       Kurtosis       2.998475