# Mean signed difference

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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.

## DefinitionEdit

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

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