Goodness-of-fit test in a multivariate errors-in-variables model $AX=B$
  We consider a multivariable functional errors-in-variables model $AX\approx B$, where the data matrices $A$ and $B$ are observed with errors, and a matrix parameter $X$ is to be estimated. A goodness-of-fit test is constructed based on the total least squares estimator. The proposed test is asymptotically chi-squared under null hypothesis. The power of the test under local alternatives is discussed. 
Alexander Kukush, Yaroslav Tsaregorodtsev
math.ST stat.TH