There seems to be a number of commercial antennas
described as H-field antennas intended for LORAN
application. Most claim improved immunity to
precipitation static. Is there a theoretical basis
for such claims?


Yes. It increases sales just like zoom zoom zoom in car advertisements.

Seriously, precipitation static is caused by corna discharge from an
antenna or object someplace near the antenna. The radiated field from
that leakage current can be almost any field impedance and will always
be a mixture of time-varying electric and magnetic fields.

What a small loop actually buys you is a compact antenna that has no
sharp protruding edges, and that decreases the chances of having corona
right from the antenna. A whip would have a sharp protruding point, and
that would encourge corona discharge and the resulting noise we call
"precipitation static".

Other than that, there is no advantage.

73 Tom