Mean signed deviation explained

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}) = (1)/(n) \sum ^{n} _{i =1} \hat{\theta _{i} } - \theta_ .

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