For a collection of distributions over a countable support set, the worst
case universal compression formulation by Shtarkov attempts to assign a
universal distribution over the support set. The formulation aims to ensure
that the universal distribution does not underestimate the probability of any
element in the support set relative to distributions in the collection. When
the alphabet is uncountable and we have a collection P of Lebesgue
continuous measures instead, we ask if there is a corresponding universal
probability density function (pdf) that does not underestimate the value of the
density function at any point in the support relative to pdfs in P.
Analogous to the worst case redundancy of a collection of distributions over
a countable alphabet, we define the \textit{attenuation} of a class to be A
when the worst case optimal universal pdf at any point x in the support is
always at least the value any pdf in the collection P assigns to x
divided by A. We analyze the attenuation of the worst optimal universal pdf
over length-n samples generated \textit{i.i.d.} from a Gaussian distribution
whose mean can be anywhere between βΞ±/2 to Ξ±/2 and variance
between Οm2β and ΟM2β. We show that this attenuation is finite,
grows with the number of samples as O(n), and also specify the
attentuation exactly without approximations. When only one parameter is allowed
to vary, we show that the attenuation grows as O(nβ), again
keeping in line with results from prior literature that fix the order of
magnitude as a factor of nβ per parameter. In addition, we also specify
the attenuation exactly without approximation when only the mean or only the
variance is allowed to vary