Efficient computation of multivariate empirical distribution functions at the observed values

David Lee () and Harry Joe
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David Lee: University of British Columbia
Harry Joe: University of British Columbia

Computational Statistics, 2018, vol. 33, issue 3, No 15, 1413-1428

Abstract: Abstract Consider the evaluation of model-based functions of cumulative distribution functions that are integrals. When the cumulative distribution function does not have a tractable form but simulation of the multivariate distribution is easily feasible, we can evaluate the integral via a Monte Carlo sample, replacing the model-based distribution function by the empirical distribution function. Given a simulation sample of size N, the naive method uses $$O(N^{2})$$ O ( N 2 ) comparisons to compute the empirical distribution function at all N sample vectors. To obtain faster computational speed when N needs to be large to achieve a desired accuracy, we propose methods modified from the popular merge sort and quicksort algorithms that preserve their average $$O(N\log _{2}N)$$ O ( N log 2 N ) complexity in the bivariate case. The modified merge sort algorithm can be extended to the computation of a d-dimensional empirical distribution function at the observed values with $$O(N\log _{2}^{d-1}N)$$ O ( N log 2 d - 1 N ) complexity. Simulation studies suggest that the proposed algorithms provide substantial time savings when N is large.

Keywords: Integral evaluation; Joint probabilities; Monte Carlo simulation; Sorting algorithms (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00180-017-0771-x

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