I am looking for statistics/research/terminology about the native language rejection by immigrants. I mean by this partial or complete refusal to use one's mother-tongue in favor of the local language (or another language suitable for communication.)
One example I can think of is early European immigrants to Israel speaking Hebrew (Ben Yehuda famously refused to speak anything but Hebrew to his family, and Amos Oz in his autobiography makes remarks about his parents' lack of intimate family vocabulary, since for them Hebrew was not the first language.)
On more anecdotal level I have heard of Algerian immigrants in France refusing to speak Arabic or Russian cold-war era immigrants to the US renouncing Russian. In this case this is likely motivated by political reasons or cultural rejection.
In a softer form this may appear as refusing to use the native language with one's children - this seems to be a relatively widespread attitude in immigrant communities, as a means of facilitating children' integration into local society.