Are you saying linear regression = statistics? Linear regression is something you learn in your first class, it's hardly ever the main model used.
And if you're going to claim that economists are publishing influential papers in other fields - and especially if you're claiming that they're doing so in an unprecedented way, with no inter-disciplinary collaboration - please provide some examples. And if you're thinking of Freakonomics, know that no researcher takes Freakonomics seriously, and neither should you.
As for sociologists "invading" economics, they sort of are. Economics and sociology have quite a bit of overlap, and researchers from the two fields often collaborate. And any group researching economic phenomena, even an inter-disciplinary one involving sociologists, would be identified as economists, not sociologists, by people reading their work. Although David Graeber, an anthropologist, did write an excellent book on economic phenomena in "Debt: the first 5000 years", and it has done quite well. You could say that it's "influential".
Unfortunately, neoclassical economics also has wide political support among the people it benefits: wealthy people and institutions, e.g. banks. Which also means they get bankrolled (hah) much more than other social scientists, which means they get preferential treatment. E.g., this very "Nobel prize" in economy that this theead is about is funded by a bank.
The force to change economics qould have to come from within economics, perhaps from behavioural economics, or new Keynesian economics (the first one seems more promising), or even from movements like degrowth or circular economics. You can't expect a sociologist to fix a different field, and that wasn't the point. The point was simply that sociology doesn't suffer this embarassment because they are not burdened by ideological pressure backed by monies interest.