Mean signed deviation explained
In statistics, the mean signed difference (MSD), also known as mean signed deviation and mean signed error, is a sample statistic that summarises 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
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
\operatorname{MSD}(\hat{\theta})=
\hat{\thetai
} - \theta_ .
Use Cases
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
\operatorname{MSD}(\hat{\thetai})=0
. 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