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Identifying Electoral Fraud: A Novel Test and New Data From Nigeria
Unformatted Document Text:  10. Indeed, only very particular (and peculiar) distributional assumptions will consistently produce last digits that are not uniformly distributed. Proposition 3 relaxes the assumption that the support of the distribution of electoral returns has to be a multiple of numeral base b, but in turn it requires that the density has to approach 0 for very large or very small returns. In reality, this is an issue only for small returns, because we cannot reasonably extend the support of f below its natural lower bound of 0. If there is a non-trivial probability of observing less than eighteen votes for a unit of interest, then proposition 3 does not hold (although proposition 1 still may). For convenience, proposition 3 explicitly imposes a restriction on the linear approximation error, which is equivalent to the restriction discussed in corollary 1. 2 Proposition 3. Consider a discrete, non-negative random variable X with probability den- sity function f and domain {s 1 , . . . , s 2 }. Suppose f can be approximated by an arithmetic progression for any sequence containing 2b − 1 elements, where b is the base of the positional numeral system, and the approximation error follows function f e , where E[f e (z + d)] = 0 over z ∈ {s 1 , . . . , s 2 −2(b−1)} for any d ∈ {0, . . . , b−1}. Then the occurrence of numerals in the last digit of X approaches a uniform distribution as f (x) approaches 0 for x ≤ s 1 + 2b − 3 and x ≥ s 2 − 2b + 3. The intuition behind the proof of proposition 3 is similar to the one for proposition 1. Here we show that the total density for different last digits in sequences of size 2(b − 1) is proportional to a constant if we can linearly approximate the density function within each sequence. In the proof of proposition 1, we broke density function g into consecutive pieces of size b. Here the pieces are overlapping, with a sequence starting at each integer, and in turn the density function’s support no longer has to be divisible by b. Finally, no formal proof is needed to see that if last digits are independently and uni- formly distributed, then (a) in expectation no last digit will be repeated more frequently than any other in a series of N random draws, and (b) the expected number of repetitions (i.e. consecutive draws of the same last digit) is N −1 b . We argue that the type of empirical data we consider lends itself to the assumption that last digits are independently distributed. It is certainly possible that the last digit of the total number of votes cast at a polling station is correlated with the last digit of the vote count at the next polling station. But if turnout is in the several hundreds, as it is in our data, it would take a spatial correlation of unlikely magnitude to carry through to the last digit. Also note that if last and penultimate digits are independently distributed, and last digits are distributed uniformly, then the expected number of pairs with digit repetition is again N −1 b , regardless of how the penultimate digit is distributed. Even if the second-to- last digit was always the same, it would not change the fact that the last digit is a match with probability 1b . If we think about the minimum distance between penultimate and last digits more generally (for convenience, we like to visualize numerals in a circle, in which case it is easy to see that the minimum distance between 7 and 1, for example, is 4), we can say that this distance is 0 with probability 1b , it is 1 with probability 2b , and it is greater than 1 with probability b−3 b (for b > 2). We later use simulations to construct confidence 2 Note that in this case f is linearly approximated over sequences of size 2(b − 1) rather than b, which means that proposition 3 actually places a somewhat stricter restriction on the approximation error than corollary 1. 5

Authors: Beber, Bernd. and Scacco, Alexandra.
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10. Indeed, only very particular (and peculiar) distributional assumptions will consistently
produce last digits that are not uniformly distributed.
Proposition 3 relaxes the assumption that the support of the distribution of electoral
returns has to be a multiple of numeral base b, but in turn it requires that the density has
to approach 0 for very large or very small returns. In reality, this is an issue only for small
returns, because we cannot reasonably extend the support of f below its natural lower bound
of 0. If there is a non-trivial probability of observing less than eighteen votes for a unit of
interest, then proposition 3 does not hold (although proposition 1 still may).
For convenience, proposition 3 explicitly imposes a restriction on the linear approximation
error, which is equivalent to the restriction discussed in corollary 1.
2
Proposition 3. Consider a discrete, non-negative random variable X with probability den-
sity function f and domain {s
1
, . . . , s
2
}. Suppose f can be approximated by an arithmetic
progression for any sequence containing 2b − 1 elements, where b is the base of the positional
numeral system, and the approximation error follows function f
e
, where E[f
e
(z + d)] = 0
over z ∈ {s
1
, . . . , s
2
−2(b−1)} for any d ∈ {0, . . . , b−1}. Then the occurrence of numerals in
the last digit of X approaches a uniform distribution as f (x) approaches 0 for x ≤ s
1
+ 2b − 3
and x ≥ s
2
− 2b + 3.
The intuition behind the proof of proposition 3 is similar to the one for proposition 1.
Here we show that the total density for different last digits in sequences of size 2(b − 1) is
proportional to a constant if we can linearly approximate the density function within each
sequence. In the proof of proposition 1, we broke density function g into consecutive pieces
of size b. Here the pieces are overlapping, with a sequence starting at each integer, and in
turn the density function’s support no longer has to be divisible by b.
Finally, no formal proof is needed to see that if last digits are independently and uni-
formly distributed, then (a) in expectation no last digit will be repeated more frequently
than any other in a series of N random draws, and (b) the expected number of repetitions
(i.e. consecutive draws of the same last digit) is
N −1
b
. We argue that the type of empirical
data we consider lends itself to the assumption that last digits are independently distributed.
It is certainly possible that the last digit of the total number of votes cast at a polling station
is correlated with the last digit of the vote count at the next polling station. But if turnout
is in the several hundreds, as it is in our data, it would take a spatial correlation of unlikely
magnitude to carry through to the last digit.
Also note that if last and penultimate digits are independently distributed, and last
digits are distributed uniformly, then the expected number of pairs with digit repetition is
again
N −1
b
, regardless of how the penultimate digit is distributed. Even if the second-to-
last digit was always the same, it would not change the fact that the last digit is a match
with probability
1
b
. If we think about the minimum distance between penultimate and last
digits more generally (for convenience, we like to visualize numerals in a circle, in which case
it is easy to see that the minimum distance between 7 and 1, for example, is 4), we can
say that this distance is 0 with probability
1
b
, it is 1 with probability
2
b
, and it is greater
than 1 with probability
b−3
b
(for b > 2). We later use simulations to construct confidence
2
Note that in this case f is linearly approximated over sequences of size 2(b − 1) rather than b, which means
that proposition 3 actually places a somewhat stricter restriction on the approximation error than corollary 1.
5


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