In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta. We then show that if the perturbation corresponds to\operatorname{rd} }\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases. When the points in this fixed set are uniformly spaced, our analysis can be viewed as a generalization of Sheppard's corrections. We discuss how our bounds can be used to balance measurement error with sample error in a rigorous way, as well as how they can be used to generalize classical numerical analysis results. The frameworks developed in our analysis can be applied to a wider range of applications than those studied in this paper and may be of general interest or (ii) rounding randomly to either the nearest larger or smaller point in this same set with probabilities proportional to the distances to these points. In both cases, we show, under reasonable assumptions on the density function of X, how to compute a constant C such that |\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]| < C \epsilon^2, provided }\operatorname{rd}[ |X^k-\operatorname{rd}(X)^{k}| ] are also given} .
We study the effects of rounding on the moments of a random variable . Specifically, given a random variable X and its rounded counterpart (X), we study | %DIFDELCMD < {\rm %%% \mathbf{E | %DIFDELCMD < {\rm %%% for non-negative integer k. We consider the case that the rounding function\operatorname{rd} : \mathbb{R}\to\mathbb{F rounding to the nearest point in some discrete set \mathbb{F or (ii) rounding randomly to either the nearest larger or smaller point in this same set with probabilities proportional to the distances to these points. In both cases, we show, under reasonable assumptions on the density function of X, how to compute a constant C such that |\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]| < C \epsilon^2, provided }\operatorname{rd}(x) - x| \leq \epsilon E(x). Asymptotic and non-asymptotic bounds for the absolute moments \mathbb{E[ |X^k-\operatorname{rd}(X)^{k}| ] are also given} .
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In this paper, we provide general bounds on the mean absolute difference and difference of
We study the effects of rounding on the
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X and a perturbation rd
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for non-negative integer k. We consider the case that the rounding function
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fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases. When the points in this fixed set are uniformly spaced, our analysis can be viewed as a generalization of Sheppard's corrections. We discuss how our bounds can be used to balance measurement error with sample error in a rigorous way, as well as how they can be used to generalize classical numerical analysis results. The frameworks developed in our analysis can be applied to a wider range of applications than those studied in this paper and may be of general interest
discrete set \mathbb{F
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