Uniformity

1
Uniformity

• serial test

• Kolmogorov-Smirnov test

2
Test

• Break up the unit interval into k bins

• If data are uniform, expect n/k in each bin

(make sure you expect at least 5 in each bin)
Uniformity
3
Test
In our sample with n100,000
k20
Uniformity
4
Test
In our sample with n100,000, k20
We passed!
Uniformity
5
Serial Test
Idea Bunch up the data into vectors.
If the individual values are independent,
the vectors should be uniformly distributed in
the unit cube.
Uniformity and Independence
6
Serial Test
Dump the data into bins of some width
Uniformity and Independence
7
Serial Test

• k by k grid

• let Oij be the number of observations in the
  i-jth bin

• if uniform and members of pairs independent,
  expect (n/2)/k2 in each bin

Uniformity and Independence
8
Serial Test
For our sample, I used a 20 by 20 grid
Of the 50,000 pairs
We expect 125 in each cell
We passed!
Uniformity and Independence
9
Serial Test
Incidentally
(failed!)
Uniformity and Independence
10
Kolmogorov-Smirnov Test
Uniformity
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Kolmogorov-Smirnov Test
Uniformity
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Kolmogorov-Smirnov Test
If X1, X2, , Xn really come from the
distribution with cdf F, the distance
should be small.
Uniformity
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Kolmogorov-Smirnov Test
Computing the test statistic
(obviously simplified)
Uniformity
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Kolmogorov-Smirnov Test
0.6 0.2 0.5 0.9 0.1 0.4 0.2
Put them in order
0.6 0.2 0.5 0.9 0.1 0.4 0.2
Now the empirical cdf is
Uniformity
15
Kolmogorov-Smirnov Test
0.6 0.2 0.5 0.9 0.1 0.4 0.2
Uniformity
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Kolmogorov-Smirnov Test
Uniformity
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Kolmogorov-Smirnov Test
0.6 0.2 0.5 0.9 0.1 0.4 0.2
Uniformity
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Kolmogorov-Smirnov Test
Let X(1), X(2), ,X(n) be the ordered sample.
Then Dn can be estimated by
This is exact for the uniform distribution!
where
Uniformity
(assuming non-repeating values)
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Kolmogorov-Smirnov Test

• We reject that this sample came from the
  proposed distribution if the empirical cdf is
  too far from the true cdf of the proposed
  distribution

• ie We reject if Dn is too large.

• ie How large is large?

Uniformity
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Kolmogorov-Smirnov Test
In the 1930s, Kolmogorov and Smirnov showed that
Uniformity
21
Kolmogorov-Smirnov Test
For small samples, people have worked out and
tabulated critical values, but there is no nice
closed form solution.

• J. Pomeranz (1973)
• J . Durbin (1968)

Uniformity
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Kolmogorov-Smirnov Test
For our small sample of size 7,
From a table, the critical value for a 0.05 level
test for n7 is 0.483.
We passed!
Uniformity
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Kolmogorov-Smirnov Test
We passed!
Uniformity