Mean signed deviation

In statistics, the '''mean signed difference''' ('''MSD'''),<ref>{{cite journal |last1=Harris |first1=D. J. |last2=Crouse |first2=J. D. |year=1993 |title=A Study of Criteria Used in Equating |journal=Applied Measurement in Education |volume=6 |issue=3 |page=203 |doi=10.1207/s15324818ame0603_3 }}</ref> also known as '''mean signed deviation''', '''mean signed error''', or '''mean bias error'''<ref>{{cite journal |last=Willmott |first=C. J. |year=1982 |title=Some Comments on the Evaluation of Model Performance |journal=Bulletin of the American Meteorological Society |volume=63 |issue=11 |page=1310|doi=10.1175/1520-0477(1982)063<1309:SCOTEO>2.0.CO;2 |bibcode=1982BAMS...63.1309W |doi-access=free }}</reF> is a sample statistic that summarizes how well a set of estimates <math>\hat{\theta}_i</math> match the quantities <math>\theta_i</math> 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 <math>\theta_i</math> would be the ''i''-th out-of-sample value of the dependent variable, and <math>\hat{\theta}_i</math> would be its predicted value. The mean signed deviation is the average value of <math>\hat{\theta}_i-\theta_i.</math>

==Definition==

The mean signed difference is derived from a set of ''n'' pairs, <math>( \hat{\theta}_i,\theta_i)</math>, where <math> \hat{\theta}_i</math> is an estimate of the parameter <math>\theta</math> in a case where it is known that <math>\theta=\theta_i</math>. In many applications, all the quantities <math>\theta_i</math> 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 <math>\hat{\theta}_i</math> being the predicted value of a series at a given lead time and <math>\theta_i</math> being the value of the series eventually observed for that time-point. The mean signed difference is defined to be :<math>\operatorname{MSD}(\hat{\theta}) = \frac{1}{n}\sum^{n}_{i=1} \hat{\theta_{i}} - \theta_{i} .</math>

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

==References== {{reflist}}

{{DEFAULTSORT:Mean Signed Difference}} Category:Summary statistics Category:Means Category:Distance