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.


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} .

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