Mean signed deviation

In statistics, the mean signed difference (MSD),^[1] also known as mean signed deviation, mean signed error, or mean bias error^[2] is a sample statistic that summarizes how well a set of estimates ${\displaystyle {\hat {\theta }}_{i}}$ match the quantities ${\displaystyle \theta _{i}}$ 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 ${\displaystyle \theta _{i}}$ would be the i-th out-of-sample value of the dependent variable, and ${\displaystyle {\hat {\theta }}_{i}}$ would be its predicted value. The mean signed deviation is the average value of ${\displaystyle {\hat {\theta }}_{i}-\theta _{i}.}$

The mean signed difference is derived from a set of n pairs, ${\displaystyle ({\hat {\theta }}_{i},\theta _{i})}$, where ${\displaystyle {\hat {\theta }}_{i}}$ is an estimate of the parameter ${\displaystyle \theta }$ in a case where it is known that ${\displaystyle \theta =\theta _{i}}$. In many applications, all the quantities ${\displaystyle \theta _{i}}$ 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 ${\displaystyle {\hat {\theta }}_{i}}$ being the predicted value of a series at a given lead time and ${\displaystyle \theta _{i}}$ being the value of the series eventually observed for that time-point. The mean signed difference is defined to be

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

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

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

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