# Fitting Gaussian distribution with indirect observations

I want to estimate the mean and variance of a gaussian random variable $X$.

The realizations of $X$, i.e. $x$, are not observable. Instead, $a$ and $b$ are observed, which are related to $X$ via

$P(x \geq a+n)= b$,

where $n$ is an unknown small noise term.

How can I fit $X$ to a gaussian distribution by using a collection of $a$ and $b$ realizations?

• What do you know about $n$?
– Tim
Commented Nov 9, 2016 at 15:42
• @Tim not much. But we can assume it is white gaussian with small variance compared to the value of b. Commented Nov 9, 2016 at 22:05
• Presumably you must have the variance of your noise term decrease as $b$ gets near 0 or 1 or you'll have probabilities that are outside (0,1) Commented Nov 10, 2016 at 1:20
• Now you have something that starts to look something like a probit model. (Another common choice would be a logit model, which would correspond to unobserved logistic noise.) Commented Nov 11, 2016 at 8:37
• This sounds like quantile regression. Commented Nov 18, 2016 at 16:04