"Someone is always going to be the favourite to be sacked," said Adkins. "If you're bottom of the league, you should be the favourite probably.
“I'm very positive... I know I'm a better manager today than when I joined Southampton”
"I'm not going to hide away from that, but what a great challenge we face."
Southampton have failed to pick up a single away point this season. Despite enjoying plenty of possession at The Hawthorns, they were undone twice by Odemwingie, who scored either side of half-time.
Saints have now conceded 28 goals in the league this season and are now five points adrift of safety, but Adkins remains positive about his side's chances of avoiding relegation after just one season in the division.
"If I don't believe we can do it, then the players aren't going to believe we can do it," said the 47-year-old, who led Saints to promotion last season.
"I don't think anybody needs to give anyone a vote of confidence. 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."
Adkins, who took over as Saints boss just over two years ago, added: "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."
Three of Southampton's next four league games are at home.
Defensive woes for the Saints
Southampton have now conceded 28 goals in their first 10 games, equalling the Premier League record set by Barnsley
They host Swansea on Saturday before travelling to Loftus Road to face fellow strugglers QPR on 17 November. They then host Newcastle and Norwich on 25 and 28 November respectively.
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