The Saints continue to occupy bottom spot after two goals from Peter Odemwingie earned West Brom a 2-0 win at the Hawthorns. They have collected only four points out of a possible 30 and conceded 28 goals - the worst defensive record in the top flight.
"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'," Adkins said. "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."
He added: "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 dropped goalkeeper Artur Boruc, who has been the subject of a club investigation into allegedly throwing a water bottle into the crowd during the recent game with Tottenham.
He said: "There is an internal investigation going on and I'd like to think that would be concluded quickly. Because of that, he has not been selected tonight."
Albion are up to fifth place in the table after bouncing back from injury-time defeats against Manchester City and Newcastle. But Baggies boss Steve Clarke is refusing to get carried away with a points tally of 17 from the first 10 games.
"I'm happy. The performances were good in the last two games but we got no points," he said. "It was just important to get back on track tonight. The performance was okay. It wasn't exceptional but okay.
"We had good control of the game, especially in the second half. We didn't look under too much threat and always looked like we could score on the counter-attack."
Source: PA
Source: PA