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Math Help - bound on parametric function

  1. #1
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    bound on parametric function

    I would like to find an upper and/or lower bound for the parametric function:
    (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 \phi(t) is the standard normal density and \Phi(t) the corresponding distribution function. The bound should be in terms of x and y, and not in t.

    To get a rough idea of the function, the attached figure shows the function for t\leq 2. bound on parametric function-figure.jpg

    Can anybody help?
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  2. #2
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    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?
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  3. #3
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    Re: bound on parametric function

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

    Quote Originally Posted by HallsofIvy View Post
    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?
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