After researching and looking up the WordPress Codex, this is the code that I have used to authenticate..
I hope this helpful to all friends .it’s working fine for me 🙂
$creds = array(
'user_login' => $username,
'user_password' => $userpassword,
'remember' => true
);
if(!empty($username) && !empty($userpassword)) {
if(!username_exists($username)) {
$userdata = array(
'user_login' => $username,
'user_email' => $useremail,
'user_pass' => $userpassword,
'role' => 'customer'
);
$user_id = wp_insert_user( $userdata ) ;
$user = wp_signon( $creds, false );
wp_set_current_user($user->ID,$username);
wp_set_auth_cookie($user->ID, true, false );
wp_redirect(site_url());
}