The age of the universe is currently estimated by astronomer to be
between 10 and 15 billion years old.  My understanding of how this
figure is arrived at is that astronomers measure the distance of the
most distantly visible objects in the universe using the Doppler
effect and the Hubble constant.

My question is this: if the most distant objects in the universe
appear to be 10-15 billion light years away, and are receding from us
at nearly the speed of light, wouldn't this mean that the universe is
actual double this age, since it would take at least 15 billion years
for an object to move 15 billion light years away, and then another 15
billion years for the light emitted at that point to reach us?

Hello.

Your question, stated another way, is addressed on Cornell
University's "Curious About Astronomy" web site.

"If light from a galaxy took 10 billion years to reach us, isn't the
galaxy farther than 10 billion light years away now?
See: Dave Rothstein, January 2003, "Ask an Astromer," hosted by
Cornell:
http://curious.astro.cornell.edu/question.php?number=436

Rothstein's key point is that "when we say that light from a galaxy
took 10 billion years to get here, we mean that the galaxy is 10
billion light years away now, and it was some distance closer than
that 10 billion years ago when the light was emitted."


Also see UCLA's Frequently Asked Questions in Cosmology:
"If the Universe is only 10 billion years old, why isn't the most
distant object we can see 5 billion light years away?
UCLA Cosmology FAQ:
http://www.astro.ucla.edu/~wright/cosmology_faq.html#ct2

search strategy: "billion years old", "billion light years away"

I hope this helps.

I am pleased with the quickness of your response.  Both sites were
very helpful and informative not only regarding my question but
related subjects.