Mean signed deviation

Summary

In statistics, the mean signed difference (MSD),[1] also known as mean signed deviation, mean signed error, or mean bias error[2] is a sample statistic that summarizes how well a set of estimates match the quantities that they are supposed to estimate. It is one of a number of statistics that can be used to assess an estimation procedure, and it would often be used in conjunction with a sample version of the mean square error.

For example, suppose a linear regression model has been estimated over a sample of data, and is then used to extrapolate predictions of the dependent variable out of sample after the out-of-sample data points have become available. Then would be the i-th out-of-sample value of the dependent variable, and would be its predicted value. The mean signed deviation is the average value of

Definition

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The mean signed difference is derived from a set of n pairs,  , where   is an estimate of the parameter   in a case where it is known that  . In many applications, all the quantities   will share a common value. When applied to forecasting in a time series analysis context, a forecasting procedure might be evaluated using the mean signed difference, with   being the predicted value of a series at a given lead time and   being the value of the series eventually observed for that time-point. The mean signed difference is defined to be

 

Use Cases

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The mean signed difference is often useful when the estimations   are biased from the true values   in a certain direction. If the estimator that produces the   values is unbiased, then  . However, if the estimations   are produced by a biased estimator, then the mean signed difference is a useful tool to understand the direction of the estimator's bias.

See also

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References

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  1. ^ Harris, D. J.; Crouse, J. D. (1993). "A Study of Criteria Used in Equating". Applied Measurement in Education. 6 (3): 203. doi:10.1207/s15324818ame0603_3.
  2. ^ Willmott, C. J. (1982). "Some Comments on the Evaluation of Model Performance". Bulletin of the American Meteorological Society. 63 (11): 1310. Bibcode:1982BAMS...63.1309W. doi:10.1175/1520-0477(1982)063<1309:SCOTEO>2.0.CO;2.