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In statistics, the mean signed difference (MSD), also known as mean signed error (MSE), is a sample statistic that summarises how well an estimator $\hat{\theta }$ matches the quantity $\theta$ that it is 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.

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

$\mathrm{MSD}(\hat{\theta })=\sum _{i=1}^n\frac{\widehat{{\theta }_i}-{\theta }_i}{n}.$

See also

• Mean absolute error

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