Mean signed deviation

In statistics, the mean signed difference (MSD), also known as mean signed deviation, mean signed error, or mean bias error is a sample statistic that summarizes 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

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

Use Cases

The mean signed difference is often useful when the estimations $\widehat{{\theta }_i}$ are biased from the true values ${\theta }_i$ in a certain direction. If the estimator that produces the $\widehat{{\theta }_i}$ values is unbiased, then $\mathrm{MSD}(\widehat{{\theta }_i})=0$. However, if the estimations $\widehat{{\theta }_i}$ 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