137,757 research outputs found
On the Uniformity of Modulo 1
It has been conjectured that the sequence modulo is uniformly
distributed. The distribution of this sequence is signifcant in relation to
unsolved problems in number theory including the Collatz conjecture. In this
paper, we describe an algorithm to compute modulo to .
We then statistically analyze its distribution. Our results strongly agree with
the hypothesis that modulo 1 is uniformly distributed.Comment: 12 pages, 2 figure
Testing multivariate uniformity based on random geometric graphs
We present new families of goodness-of-fit tests of uniformity on a
full-dimensional set based on statistics related to edge lengths
of random geometric graphs. Asymptotic normality of these statistics is proven
under the null hypothesis as well as under fixed alternatives. The derived
tests are consistent and their behaviour for some contiguous alternatives can
be controlled. A simulation study suggests that the procedures can compete with
or are better than established goodness-of-fit tests. We show with a real data
example that the new tests can detect non-uniformity of a small sample data
set, where most of the competitors fail.Comment: 36 pages, 2 figure
Le Cam spacings theorem in dimension two
The definition of spacings associated to a sequence of random variables is
extended to the case of random vectors in [0,1]^2. Beirlant & al. (1991) give
an alternative proof of the Le Cam (1958) theorem concerning asymptotic
normality of additive functions of uniform spacings in [0,1]. I adapt their
technique to the two-dimensional case, leading the way to new directions in the
domain of Complete Spatial Randomness (CSR) testing
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