A New Method for Bounding the Distance Between Sums of Independent Integer-Valued Random Variables |
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Authors: | Eutichia Vaggelatou |
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Institution: | 1.Section of Statistics and OR, Department of Mathematics,University of Athens,Athens,Greece |
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Abstract: | Let X 1, X 2,..., X n and Y 1, Y 2,..., Y n be two sequences of independent random variables which take values in ? and have finite second moments. Using a new probabilistic method, upper bounds for the Kolmogorov and total variation distances between the distributions of the sums \(\sum_{i=1}^{n}X_{i}\) and \(\sum_{i=1}^{n}Y_{i}\) are proposed. These bounds adopt a simple closed form when the distributions of the coordinates are compared with respect to the convex order. Moreover, they include a factor which depends on the smoothness of the distribution of the sum of the X i ’s or Y i ’s, in that way leading to sharp approximation error estimates, under appropriate conditions for the distribution parameters. Finally, specific examples, concerning approximation bounds for various discrete distributions, are presented for illustration. |
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