$\vB_{_{\rm MLE}} $ which renders a minimum in $$\chi^2=\frac{1}{\ssigint}\sum_{cz_i <r}
{\left(s M_{0i}+ 2.17s \frac{  \vB \cdot \hat {\vr}_i}{ cz_i}+\eta_0-\eta_i\right)^2}\;$$ with respect to the three components of $\vB$. The sum is over galaxies within $r$ and $\hat {\vr}_i$ is unit vector in the direction of galaxy $i$. Further, in this expression we have approximated $P=-5{\rm log}(1-\vB \cdot \hat {\vr}_i/cz_i)\approx
 2.17  \vB \cdot \hat {\vr}_i/{ cz_i}$.

Tests {#sec:test}
=====

In order to test the performance of  and the MLE reconstructions of the bulk flow from the SFI++ TF data we use 2200 mock catalogs of TF measurements. In each of the catalogues, galaxies with the same positions as in the real SFI++ data are assigned absolute magnitudes, $M_i$, and line width parameters, $\eta_i$, following an artificial ITF relation with slope $s=-0.12$ and intrinsic scatter $ \sigma_\eta=0.057$ [e.g. @DN10]. The peculiar velocities of galaxies in each mock are taken from a linear gaussian random velocity field in a cubic box of $1454\hmpc$ on the side. Each mock is placed randomly in this large box and the peculiar velocity of each galaxy is then obtained by interpolating the velocity field on the position of the galaxy. A gaussian random realizaton of the velocity field 