Normalizing/standardizing scores with iterative estimates of mean and standard deviation

Hello Statistics lovers,

I'm trying to find an expression for the distribution of scores (from a
normal/gaussian distribution) which are normalized/standardized using
iterative estimates (EWMA/EWMV) of the mean and standard deviation.

What I am looking for is an *analog* of the way the T distribution
specifies the distribution of scores (from a normal/gaussian
distribution) which are normalized/standardized using a sample mean and
standard deviation from n samples. That is, t = (x - Mn) / Sn, (where x
are new incoming scores, and Mn and Sn are the sample mean and standard
deviation taken from n previous samples) is distributed by the T(n-1)
distribution.

For the problem I am interested in, the "iterative estimates" are
obtained as follows:

1. The iterative mean (IM(i) after incoming score x(i)) is effectively
the EWMA (exponentially weighted moving average), computed as such:

IM(i) = lambda * x(i) + (lambda - 1) * IM(i-1)

2. The iterative standard deviation(IS(i) after incoming score x(i)) is
effectively the EWMV (exponentially weighted moving variance), computed
as such:

IS(i)^2 = alpha * (x(i) - IM(i-1))^2 + (alpha - 1) * IS(i-1)^2

(Note: I've also seen these expressed with lambda <-> 1 - lambda, and
alpha <-> 1 - alpha. Also, often alpha := lambda)

So, the problem is then to find the distribution of the
normalized/standardized scores:
y(i) = (x(i) - IM(i-1)) / IS(i-1)
as a function of lambda and alpha, and of course the underlying
population mean and standard deviation.
Numerically approximating this distribution is not a problem, but I
would like to find an analytical expression.

I have found plenty of references to the distribution of the EWMA from
the field of statistical process control (SPC); however this doesn't
quite address what I'm looking for.
Otherwise, I haven't had any real luck tracking this down. Perhaps I'm
using the wrong search terms. Has anyone come across this problem
before ?

Thanks in advance,
Joe

.

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