I have an experiment in which I am taking measurements of a normally distributed variable $Y$,
$$Y \sim N(\mu,\sigma)$$
However, previous experiments have provided some evidence that the standard deviation $\sigma$ is an affine function of an independent variable $X$, i.e.
$$\sigma = a|X| + b$$
$$Y \sim N(\mu,a|X| + b)$$
I would like to estimate the parameters $a$ and $b$ by sampling $Y$ at multiple values of $X$. Additionally, due to experiment limitations I can only take a limited (roughly 30-40) number of samples of $Y$, and would prefer to sample at several values of $X$ for unrelated experimental reasons. Given these constraints, what methods are available to estimate $a$ and $b$?
This is extra information, if you're interested in why I'm asking the above question. My experiment measures auditory and visual spatial perception. I have an experiment setup in which I can present either auditory or visual targets from different locations, $X$, and subjects indicate the perceived location of the target, $Y$. Both vision* and audition get less precise with increasing eccentricity (i.e. increasing $|X|$), which I model as $\sigma$ above. Ultimately, I'd like to estimate $a$ and $b$ for both vision and audition, so I know the precision of each sense across a range of locations in space. These estimates will be used to predict the relative weighting of visual and auditory targets when presented concurrently (similar to the theory of multisensory integration presented here: http://www.ncbi.nlm.nih.gov/pubmed/12868643).
*I know that this model is inaccurate for vision when comparing foveal to extrafoveal space, but my measurements are constrained solely to extrafoveal space, where this is a decent approximation.