The article “On the Statistical Analysis of Rounded or Binned Data” discusses the impact of rounding or binning on statistical analyses. It explores Sheppard’s corrections and the total variation bounds on the rounding error in estimating the mean. It also introduces bounds based on Fisher information. The article highlights the importance of addressing errors when computing statistical values from rounded or binned data.
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Statistical Analysis of Rounded or Binned Data
Sheppard’s corrections offer approximations, but errors persist. Analytical bounds provide insight into the magnitude of these errors.
Imagine having a list of length measurements in inches, precise to the inch. Our goal is to estimate the average height within this cohort.
Sheppard’s corrections
Sheppard’s corrections can be applied to estimate original data from rounded values. The corrections are useful when the probability density function is sufficiently smooth, the bin width is not too large, and the sample size is moderate.
Total variation bounds on the rounding error in estimating the mean
By considering the total variation of the probability density function, we can constrain the error in computing the mean based on rounded or binned data. Practical upper bounds for the rounding error can be expressed in terms of the Fisher information and higher derivatives of the probability density function.
Bounds based on Fisher information
Additional bounds on the rounding error when estimating the mean can be expressed in terms of the Fisher information and higher derivatives of the probability density function of the unknown distribution.
Conclusion
Sheppard’s corrections are approximations that can be used to account for errors in computing the mean, variance, and other moments of a distribution based on rounded or binned data. Additional bounds on the rounding error when estimating the mean can be expressed in terms of the Fisher information and higher derivatives of the probability density function of the unknown distribution.
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