I am trying to carry out a simple linear regression analysis represented by the standard formulation: $$ y_i = \beta_0 + \beta_1x_i +\epsilon_i $$
However, there is an issue with my data. There is no guarantee that the response variable values are the true values and the probability of it being true decreases as the value of $y_i$ increases, i.e., the higher the value of $y_i$, the less we are certain that $y_i$ is the true value.
I am trying to deal with this using a weighted least square regression where I assign weights $1/y_i$ to each $y_i$ to represent the uncertainty of $y_i$.
There are certainly other ways of assigning weights but I am not sure if this kind of procedure makes any sense. Moreover, if the resulting model is significant, does it actually suggest there is a relationship between $x$ and $y$?
Thanks very much!
P.S. I think I need to provide a bit more of context. The data I have is derived, both $y$ and $x$ where $y$ is the nearest distance between two point patterns $A$ and $B$. From prior knowledge, $B$ is generated by $A$ but I do not know which point in $B$ is generated by which point in $A$. So I have to make assumption that the closer the point $B$ is to $A$ then point $B$ is generated by $A$. $x$ is some other quantity associated with $A$. So now the problem is that the farther away a certain point $B$ is from its nearest neighbor in $A$, the less certain I am that this point in $B$ is actually generated by that point in $A$. So in some way I am trying to penalize the validity of large values of $y$.