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 θ ^ i {\displaystyle {\hat {\theta }}_{i}} match the quantities θ i {\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 θ i {\displaystyle \theta _{i}} would be the i-th out-of-sample value of the dependent variable, and θ ^ i {\displaystyle {\hat {\theta }}_{i}} would be its predicted value. The mean signed deviation is the average value of θ ^ i − θ i . {\displaystyle {\hat {\theta }}_{i}-\theta _{i}.}

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

MSD ⁡ ( θ ^ ) = 1 n ∑ i = 1 n θ i ^ − θ i . {\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 θ i ^ {\displaystyle {\hat {\theta _{i}}}} are biased from the true values θ i {\displaystyle \theta _{i}} in a certain direction. If the estimator that produces the θ i ^ {\displaystyle {\hat {\theta _{i}}}} values is unbiased, then MSD ⁡ ( θ i ^ ) = 0 {\displaystyle \operatorname {MSD} ({\hat {\theta _{i}}})=0} . However, if the estimations θ i ^ {\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