No, they are not. It's fundamentally impossible to secure proprietary software because you have to trust its provider the software does what it says it does whereas with open-source you can always check for yourself. Any backdoor in open-source software is there to be exposed and corrected.
With proprietary software only one party can disclose vulnerabilities and in open-source anyone with the knowledge can do it. You can choose to trust a single party or choose to trust a myriad of different parties any one of which can blow the whistle if they find something fishy.
I find it highly unlikely a backdoor to a popular open-source application could remain there for long. I don't think it's unlikely at all with proprietary software where there is no incentive to fix a problem until someone outside the company learns about it.
True, but there is no way to prove it's secure. It's not about convincing myself or anyone else - it's about proof.
But maybe we can agree on the following. Closed source software can be secure but there is a broad spectrum of needs for convincing someone that a software is secure and this need may be better served with open source software in some circumstances. For some it is sufficient to trust a vendor. Some want to audit the source code (and this does not exclude closed source software). Some even need formal verification maybe even of the underlying hardware.
http://www.theregister.co.uk/2009/08/14/critical_linux_bug/
http://www.networkworld.com/community/blog/linux-finally-fix...
http://it.slashdot.org/story/11/06/20/2257229/13-year-old-pa...
You're trying to apply an impossible standard to closed source software that software that was that developed from the start to be open source and developed in the open cannot meet.