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.

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

References

Text derived from https://en.wikipedia.org/w/index.php?oldid=1259440311 under CC BY-SA 4.0; modifications have been made.