Identifiability of Normal From Conditional Measure (Monotonicity of CDF Ratios)

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Let $Z_x sim mathcalN(x,1)$, $D_1 = [0,c]$, and $D=[-c,c]$. Can we determine $x$ from
$$f(x) = mathbbP(Z_xin D_1 | Z_xin D) = fracPhi(c - x) - Phi(-x)Phi(c - x) - Phi(-c-x)?$$
In particular, can we validate the (numerically obvious) claim that $f$ is monotone, ranging from $0$ to $1$? Even $lim_xtoinftyf(x) = 1$ doesn't seem obvious to me; L'Hospital's rule isn't illuminating there.



A clear approach to this is to consider the derivative
$$
beginalign*
f'(x) &= fracf(x)bigl(phi(c-x)-phi(-c-x)bigr) - bigl(phi(c-x) - phi(-x)bigr)mathbbP(Z_xin D)\
&propto f(x)bigl(phi(c-x)-phi(-c-x)bigr) - bigl(phi(c-x) - phi(-x)bigr),
endalign*
$$
and show that $f'>0$ uniformly, but I can't seem to bound this either. Answers to either would be extremely helpful, but injectivity of $f$ is more important for my application.




real-analysis probability monotone-functions upper-lower-bounds




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    up vote
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    down vote

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    Let $Z_x sim mathcalN(x,1)$, $D_1 = [0,c]$, and $D=[-c,c]$. Can we determine $x$ from
    $$f(x) = mathbbP(Z_xin D_1 | Z_xin D) = fracPhi(c - x) - Phi(-x)Phi(c - x) - Phi(-c-x)?$$
    In particular, can we validate the (numerically obvious) claim that $f$ is monotone, ranging from $0$ to $1$? Even $lim_xtoinftyf(x) = 1$ doesn't seem obvious to me; L'Hospital's rule isn't illuminating there.



    A clear approach to this is to consider the derivative
    $$
    beginalign*
    f'(x) &= fracf(x)bigl(phi(c-x)-phi(-c-x)bigr) - bigl(phi(c-x) - phi(-x)bigr)mathbbP(Z_xin D)\
    &propto f(x)bigl(phi(c-x)-phi(-c-x)bigr) - bigl(phi(c-x) - phi(-x)bigr),
    endalign*
    $$
    and show that $f'>0$ uniformly, but I can't seem to bound this either. Answers to either would be extremely helpful, but injectivity of $f$ is more important for my application.




    real-analysis probability monotone-functions upper-lower-bounds




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $Z_x sim mathcalN(x,1)$, $D_1 = [0,c]$, and $D=[-c,c]$. Can we determine $x$ from
      $$f(x) = mathbbP(Z_xin D_1 | Z_xin D) = fracPhi(c - x) - Phi(-x)Phi(c - x) - Phi(-c-x)?$$
      In particular, can we validate the (numerically obvious) claim that $f$ is monotone, ranging from $0$ to $1$? Even $lim_xtoinftyf(x) = 1$ doesn't seem obvious to me; L'Hospital's rule isn't illuminating there.



      A clear approach to this is to consider the derivative
      $$
      beginalign*
      f'(x) &= fracf(x)bigl(phi(c-x)-phi(-c-x)bigr) - bigl(phi(c-x) - phi(-x)bigr)mathbbP(Z_xin D)\
      &propto f(x)bigl(phi(c-x)-phi(-c-x)bigr) - bigl(phi(c-x) - phi(-x)bigr),
      endalign*
      $$
      and show that $f'>0$ uniformly, but I can't seem to bound this either. Answers to either would be extremely helpful, but injectivity of $f$ is more important for my application.




      real-analysis probability monotone-functions upper-lower-bounds




      share|cite|improve this question











      Let $Z_x sim mathcalN(x,1)$, $D_1 = [0,c]$, and $D=[-c,c]$. Can we determine $x$ from
      $$f(x) = mathbbP(Z_xin D_1 | Z_xin D) = fracPhi(c - x) - Phi(-x)Phi(c - x) - Phi(-c-x)?$$
      In particular, can we validate the (numerically obvious) claim that $f$ is monotone, ranging from $0$ to $1$? Even $lim_xtoinftyf(x) = 1$ doesn't seem obvious to me; L'Hospital's rule isn't illuminating there.



      A clear approach to this is to consider the derivative
      $$
      beginalign*
      f'(x) &= fracf(x)bigl(phi(c-x)-phi(-c-x)bigr) - bigl(phi(c-x) - phi(-x)bigr)mathbbP(Z_xin D)\
      &propto f(x)bigl(phi(c-x)-phi(-c-x)bigr) - bigl(phi(c-x) - phi(-x)bigr),
      endalign*
      $$
      and show that $f'>0$ uniformly, but I can't seem to bound this either. Answers to either would be extremely helpful, but injectivity of $f$ is more important for my application.





      real-analysis probability monotone-functions upper-lower-bounds





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      asked Aug 6 at 5:56









      cdipaolo

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