Mean signed deviation explained

In statistics, the mean signed difference (MSD),^[1] also known as mean signed deviation, mean signed error, or mean bias error^[2]

Notes and References

Harris . D. J. . Crouse . J. D. . 1993 . A Study of Criteria Used in Equating . Applied Measurement in Education . 6 . 3 . 203 . 10.1207/s15324818ame0603_3 .
Willmott . C. J. . 1982 . Some Comments on the Evaluation of Model Performance . Bulletin of the American Meteorological Society . 63 . 11 . 1310. 10.1175/1520-0477(1982)063<1309:SCOTEO>2.0.CO;2 . 1982BAMS...63.1309W . free . is a sample statistic that summarizes how well a set of estimates
\hat{\theta}_i

match the quantities
\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

\theta_i

would be the i-th out-of-sample value of the dependent variable, and
\hat{\theta}_i

would be its predicted value. The mean signed deviation is the average value of
\hat{\theta}_i-\theta_i.

Definition

The mean signed difference is derived from a set of n pairs,

(\hat{\theta}_i,\theta_i)

, where
\hat{\theta}_i

is an estimate of the parameter
\theta

in a case where it is known that
\theta=\theta_i

. In many applications, all the quantities
\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
\hat{\theta}_i

being the predicted value of a series at a given lead time and
\theta_i

being the value of the series eventually observed for that time-point. The mean signed difference is defined to be
\operatorname{MSD}(\hat{\theta})=

1
n
n
\sum
i=1

\hat{\theta_i

} - \theta_ .
Use Cases

The mean signed difference is often useful when the estimations

\hat{\theta_i}

are biased from the true values
\theta_i

in a certain direction. If the estimator that produces the
\hat{\theta_i}

values is unbiased, then
\operatorname{MSD}(\hat{\theta_i})=0

. However, if the estimations
\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

	n
\sum
	i=1

Mean signed deviation explained

Notes and References

Definition

Use Cases

See also

References