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Abstract:
One-dimensional Halton sequences are generated through a sequence of individual cycles that each progressively fill the 0-1 space. For a Halton sequence based on prime r, the first such cycle will be of length r-1, with subsequent cycles being of length r. It is clear that for most choices of N, the length of the sequence, and r, the prime number used, the final cycle in the sequence will not be a complete cycle. Due to the way the cycles sequentially fill the 0-1 space, this will lead to a dissymmetry in the final cycle. Depending on the relative weight of r with respect to N, this single cycle dissymmetry can lead to dissymmetry in the entire Halton sequence. This dissymmetry is then carried over into any transforms of this sequence (e.g. using an inverse normal transform), leading to draws that follow a distribution that has the wrong mean and is potentially skewed. This can have serious impacts on the efficiency of any simulation method based on these draws.
We give a detailed account of the problems caused by this dissymmetry, both in the original Halton sequences and in transforms of these sequences. We develop an approach that manages to visibly alleviate these problems with dissymmetry and show how this approach can lead to improvements in efficiency when the draws are used in simulation-based estimation.
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