# Linear Regression with heavy tailed noise

The model is linear $y_i = a\cdot x_i + b + e_i,~ i = 1,2,\ldots,N$. It is given that the noise is heavy tailed. However the distribution of noise conditional on $x$ is the same for all data points. My question is that how should I model the data generating process? Should I use a Student-t distribution for the noise process? Should I use M estimator in R? Facts: 1. OLS is not to be used. 2. 2. The noise distribution can depend on $x$, but is independent across samples. 3. conditioned on the value of $x$, noise's mean is 0. More clarification: the distribution of noise at $x$, i.e., $N(x)$ is a function of $x$. However for different data samples $i=1,2,\ldots,M$, the distribution of noise samples remains the same.

• Is this an empirical problem (i.e. fitted model with assumptions being violated) or a theoretical (math) problem?
– Jon
Commented Sep 21, 2016 at 23:44
• I am given a data set containing $x_i,y_i$ and I have to estimate $a,b$ Commented Sep 22, 2016 at 0:50
• Commented Feb 24, 2017 at 8:20

2. Create a single dataset combining squared residuals/errors, $e_i^2$ and their respective $x$ values.
3. Fit $e_i^2 = a\cdot x_i + b$. If the noise is zero mean, $e_i^2$ is equal to the variance of the error at $x_i$.