Mean signed deviation

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 ${\hat {\theta }}_{i}$ 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 ${\hat {\theta }}_{i}$ would be its predicted value. The mean signed deviation is the average value of ${\hat {\theta }}_{i}-\theta _{i}.$

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

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

Use Cases

The mean signed difference is often useful when the estimations ${\hat {\theta _{i}}}$ are biased from the true values $\theta _{i}$ in a certain direction. If the estimator that produces the ${\hat {\theta _{i}}}$ values is unbiased, then $\operatorname {MSD} ({\hat {\theta _{i}}})=0$. However, if the estimations ${\hat {\theta _{i}}}$ are produced by a biased estimator, then the mean signed difference is a useful tool to understand the direction of the estimator's bias.

See also

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