I don't know what you mean by a bound on points in the plane. Do you want bounds on x and y separately or a bound on the distance to the origin?
I would like to find an upper and/or lower bound for the parametric function:
where is the standard normal density and the corresponding distribution function. The bound should be in terms of and , and not in .
To get a rough idea of the function, the attached figure shows the function for .
Can anybody help?
I just want to find functions and such that for all :
Alternatively, having functions and such that for all :
would also be fine.
In any case the parametric function given above should be enclosed by the functions that provide the bounds. For instance, this paper provides in (2) such bounds for Mill's ratio, but my problem is for a parametric function.