Mean signed deviation
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In statistics, the mean signed difference (MSD),cite-ref-1[1] also known as mean signed deviation, mean signed error, or mean bias errorcite-ref-2[2] is a sample statistic that summarizes how well a set of estimates θ θ ^ ^ i {\displaystyle {\hat {\theta }}_{i}} match the quantities θ 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 θ i {\displaystyle \theta _{i}} would be the i-th out-of-sample value of the dependent variable, and θ θ ^ ^ i {\displaystyle {\hat {\theta }}_{i}} would be its predicted value. The mean signed deviation is the average value of θ θ ^ ^ i − − θ θ i . {\displaystyle {\hat {\theta }}_{i}-\theta _{i}.}
Contents
• See also
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Definition
The mean signed difference is derived from a set of n pairs, ( θ θ ^ ^ i , θ θ i ) {\displaystyle ({\hat {\theta }}_{i},\theta _{i})} , where θ θ ^ ^ i {\displaystyle {\hat {\theta }}_{i}} is an estimate of the parameter θ θ {\displaystyle \theta } in a case where it is known that θ θ = θ θ i {\displaystyle \theta =\theta _{i}} . In many applications, all the quantities θ 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 θ ^ ^ i {\displaystyle {\hat {\theta }}_{i}} being the predicted value of a series at a given lead time and θ 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 θ ^ ^ ) = 1 n ∑ ∑ i = 1 n θ θ i ^ ^ − − θ θ i . {\displaystyle \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 θ θ i ^ ^ {\displaystyle {\hat {\theta _{i}}}} are biased from the true values θ θ i {\displaystyle \theta _{i}} in a certain direction. If the estimator that produces the θ θ i ^ ^ {\displaystyle {\hat {\theta _{i}}}} values is unbiased, then MSD ( θ θ i ^ ^ ) = 0 {\displaystyle \operatorname {MSD} ({\hat {\theta _{i}}})=0} . However, if the estimations θ 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.
See also
References