[Exim] Aliases not recognised

This may be slightly off topic - so I apologise in advance if that is the
case.

I have an alias eximlogs@??? which points to three of our admins,
I've written a simple script which runs the daily logs through exim stats,
rotates the logs and emails the lot to eximlogs@???. The script
is:

/usr/local/exim/bin/eximstats /usr/local/exim/log/exim_mainlog
>/tmp/mail_stats.txt
/bin/mail -s "Bravo Mail Statistics" eximlogs@???
</tmp/mail_stats.txt
rm /tmp/mail_stats.txt
/usr/sbin/logrotate /etc/logrotate.conf

That is set to run by cron daily.

If I send email to eximlogs@??? through any mail server it gets
through, but when it's sent by cron (root) as the local machine I get:

eximlogs@???... User unknown
/root/dead.letter... Saved message in /root/dead.letter

"User unknown" - what a blatant lie! The output from exim -bt -d9 is:

> eximlogs@???

tonymail@???
    <-- tonyalias@???
    <-- tony@???
    <-- eximlogs@???
  deliver to tonymail in domain bravo.whitburn.xcalibre.co.uk
  director = localuser, transport = local_delivery
pammail@???
    <-- pamalias@???
    <-- pam@???
    <-- eximlogs@???
  deliver to pammail in domain bravo.whitburn.xcalibre.co.uk
  director = localuser, transport = local_delivery
tony@???
    <-- tony@???
    <-- tonyalias@???
    <-- tony@???
    <-- eximlogs@???
  deliver to tony@???
  router = lookuphost, transport = remote_smtp
  host exchange.xcalibre.co.uk [194.201.48.27] 
pam@???
    <-- pam@???
    <-- pamalias@???
    <-- pam@???
    <-- eximlogs@???
  deliver to pam@???
  router = lookuphost, transport = remote_smtp
  host exchange.xcalibre.co.uk [194.201.48.27] 
pam@???
    <-- pam@???
    <-- pamalias@???
    <-- pam@???
    <-- eximlogs@???
  deliver to pam@???
  router = lookuphost, transport = remote_smtp
  host exchange.xcalibre.co.uk [194.201.48.27] 
o.cook@???
    <-- eximlogs@???
  deliver to o.cook@???
  router = lookuphost, transport = remote_smtp
  host mail0.etoncollege.org.uk [195.195.166.9]   MX=10
  host mail1.etoncollege.org.uk [195.195.167.130] MX=25
  host mh1.rhbnc.ac.uk          [134.219.102.162] MX=40
o.cook@???
    <-- eximlogs@???
  deliver to o.cook@???
  router = lookuphost, transport = remote_smtp
  host mail0.etoncollege.org.uk [195.195.166.9]   MX=10
  host mail1.etoncollege.org.uk [195.195.167.130] MX=25
  host mh1.rhbnc.ac.uk          [134.219.102.162] MX=40

Does anyone have any ideas why this should be the case, or what I can do to
remedy it? If you need any further information, please ask!

Ollie