# Thread: bound on parametric function

1. ## bound on parametric function

I would like to find an upper and/or lower bound for the parametric function:
$\displaystyle (x,y)=\left\{ \left$$\frac{\Phi(t)}{\phi(t)}, \frac{\Phi(t)}{\phi(t)}\Phi(-t) + t\Phi(t) +\phi(t) \right$$ : -\infty\leq t \leq \infty\right\}$
where $\displaystyle \phi(t)$ is the standard normal density and $\displaystyle \Phi(t)$ the corresponding distribution function. The bound should be in terms of $\displaystyle x$ and $\displaystyle y$, and not in $\displaystyle t$.

To get a rough idea of the function, the attached figure shows the function for $\displaystyle t\leq 2$.

Can anybody help?

2. ## Re: bound on parametric function

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?

3. ## Re: bound on parametric function

I just want to find functions $\displaystyle y_l(x)$ and $\displaystyle y_h(x)$ such that for all $\displaystyle t$:
$\displaystyle y_l(x(t)) \leq y(t) \leq y_h(x(t))$
Alternatively, having functions $\displaystyle x_l(y)$ and $\displaystyle x_h(x)$ such that for all $\displaystyle t$:
$\displaystyle x_l(y(t)) \leq x(t) \leq x_h(y(t))$
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.

Originally Posted by HallsofIvy
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?