The Saints continue to occupy bottom spot after two goals from Peter Odemwingie earned West Brom a 2-0 win at the Hawthorns last night.
They have collected only four points out of a possible 30 and conceded 28 goals - the worst defensive record in the top flight.
Adkins said: "I don't think anybody needs to give anyone a vote of confidence because if you do, then we all turn around and say, 'he's got the dreaded vote of confidence'.
"We need to win and I believe we can do it. I'm very proud and privileged to be the Southampton manager. I'll stand out and lead from the front as I've always done.
"I'm very positive. I know I'm a better manager today than when I joined Southampton. I know I've improved a lot.
"But someone is always going to be the favourite [to be sacked], and if you're bottom of the league, well, you should be the favourite probably.
"I'm not going to hide away from that. But what a great challenge we face."
Adkins insists he will remain positive about the battle ahead to survive in the top flight.
He said: "We're going to embrace it with positivity. If I don't believe we can do it, then the players aren't going to believe we can do it.
"We've played eight of the top 10. I'm looking forward to the fixtures we've got coming up."