Mean signed difference

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
 * $$\operatorname{MSD}(\hat{\theta}) = \sum^{n}_{i=1} \frac{\hat{\theta_{i}} - \theta_{i}}{n} .$$