Do you believe Linux beats Windows on security?

No, Linux does not beat Windows on security and vice versa.  Both the operating systems come with their own share of vulnerabilities and features. 

In general, security is a practice/process and not something that is default to a brand or type of operating system.

Active Directory is an excellent feature of Windows that integrates identification, authentication, authorization, access control, and a lot more security into a centrally manageable database.  Linux community is yet to catch up with something similar and tightly integrated into the architecture.  However, the same Active Directory, if not configured properly can allow permissions due to leakage (inheritance due to poor configuration) to unauthorized personnel.  Linux has security that has been tested through the ages on *NIX and clones.  "shadow", NIS, LDAP, MIT Kerberos 5, etc are all configurable on Linux.  But, again the  security and assurance of the whole system depends on the ability of the person configuring the same. All default configurations as available for services in Windows and Linux are both vulnerable.  It requires a bit of hardening on the same to secure the services.

I hope I have answered your question adequately and also that this would help you in some way.