mean signed difference

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 $\hat_i$ match the quantities $\theta_i$ 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 $\theta_i$ would be the ''i''-th out-of-sample value of the dependent variable, and $\hat_i$ would be its predicted value. The mean signed deviation is the average value of $\hat_i-\theta_i.$

Definition

The mean signed difference is derived from a set of ''n'' pairs, $( \hat_i,\theta_i)$, where $\hat_i$ 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 $\hat_i$ 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
:$\operatorname(\hat) = \frac\sum^_ \hat - \theta_ .$

See also

*Bias of an estimator
*Deviation (statistics)
* Mean absolute difference
*Mean absolute error

Summary statistics
Means
Distance

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