Misplaced Pages

Mean signed deviation

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Mean signed difference)

In statistics, the mean signed difference (MSD), also known as mean signed deviation, mean signed error, or mean bias error is a sample statistic that summarizes how well a set of estimates θ ^ i {\displaystyle {\hat {\theta }}_{i}} match the quantities θ i {\displaystyle \theta _{i}} 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 θ i {\displaystyle \theta _{i}} would be the i-th out-of-sample value of the dependent variable, and θ ^ i {\displaystyle {\hat {\theta }}_{i}} would be its predicted value. The mean signed deviation is the average value of θ ^ i − θ i . {\displaystyle {\hat {\theta }}_{i}-\theta _{i}.}

Definition

The mean signed difference is derived from a set of n pairs, ( θ ^ i , θ i ) {\displaystyle ({\hat {\theta }}_{i},\theta _{i})} , where θ ^ i {\displaystyle {\hat {\theta }}_{i}} is an estimate of the parameter θ {\displaystyle \theta } in a case where it is known that θ = θ i {\displaystyle \theta =\theta _{i}} . In many applications, all the quantities θ i {\displaystyle \theta _{i}} 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 θ ^ i {\displaystyle {\hat {\theta }}_{i}} being the predicted value of a series at a given lead time and θ i {\displaystyle \theta _{i}} being the value of the series eventually observed for that time-point. The mean signed difference is defined to be

MSD ⁡ ( θ ^ ) = 1 n ∑ i = 1 n θ i ^ − θ i . {\displaystyle \operatorname {MSD} ({\hat {\theta }})={\frac {1}{n}}\sum _{i=1}^{n}{\hat {\theta _{i}}}-\theta _{i}.}

Use Cases

The mean signed difference is often useful when the estimations θ i ^ {\displaystyle {\hat {\theta _{i}}}} are biased from the true values θ i {\displaystyle \theta _{i}} in a certain direction. If the estimator that produces the θ i ^ {\displaystyle {\hat {\theta _{i}}}} values is unbiased, then MSD ⁡ ( θ i ^ ) = 0 {\displaystyle \operatorname {MSD} ({\hat {\theta _{i}}})=0} . However, if the estimations θ i ^ {\displaystyle {\hat {\theta _{i}}}} 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


References

  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.


Stub icon

This statistics-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: