Bill,

My experience with many shapes of top hats is that it DOES reduce the
radiation measured in the far field. The theoretical explanation I have

seen

is that the effective height is reduced due to the current foldback.

I have an LF antenna that uses a telescopic mast for one part of multiple
top loading (three masts in all).  The mast is parked in the down position,
and cranked up for LF transmitting.  The far field is definitely lower (from
S meter reports of several others) when the wire is below the "full" height.

The situation may well depend on how good the ground conductivity is, as a
"drooping" part of top loading means higher field strength to ground over
the droopy part, and higher field strength means higher ground current and
consequent I squared R losses.  In other words, RF warm spots can arise
under drooping wires.  If the antenna is built over an immaculate ground
system with hundreds of ground radials (as per a text book design), then the
likes of an umbrella antenna would not be as lossy as one built over average
soil with perhaps a few radials.

73, Bob ZL2CA