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Hoeffding inequality for the difference of two sample means?

  1. Aug 30, 2012 #1
    In W. Hoeffding's 1963 paper* he gives the well known inequality:

    [itex]P(\bar{x}-\mathrm{E}[x_i] \geq t) \leq \exp(-2t^2n) \ \ \ \ \ \ (1)[/itex],

    where [itex]\bar{x} = \frac{1}{n}\sum_{i=1}^nx_i[/itex], [itex]x_i\in[0,1][/itex]. [itex]x_i[/itex]'s are independent.

    Following this theorem he gives a corollary for the difference of two sample means as:

    [itex]P(\bar{x}-\bar{y}-(\mathrm{E}[x_i] - \mathrm{E}[y_k]) \geq t) \leq \exp(\frac{-2t^2}{m^{-1}+n^{-1}}) \ \ \ \ \ \ (2)[/itex],

    where [itex]\bar{x} = \frac{1}{n}\sum_{i=1}^nx_i[/itex], [itex]\bar{y} = \frac{1}{m}\sum_{k=1}^my_k[/itex], [itex]x_i,y_k\in[0,1][/itex]. [itex]x_i[/itex]'s and [itex]y_k[/itex]'s are independent.


    My question is: How does (2) follow from (1)?

    -Jan

    *http://www.csee.umbc.edu/~lomonaco/f08/643/hwk643/Hoeffding.pdf (equations (2.6) and (2.7))
     
  2. jcsd
  3. Aug 30, 2012 #2

    chiro

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    Hey JanO and welcome to the forums.

    One idea I have is to let Z = X + Y and use Z instead of X in the definition.
     
  4. Aug 30, 2012 #3
    Thanks Chiro for your responce.

    However, I still do not understand how the term [itex](m^{-1} + n^{-1})[/itex] comes into the bound. Isn't [itex]z=\bar{x}-\bar{y}[/itex] is still bounded between [0,1]?

    -Jan
     
  5. Aug 30, 2012 #4

    chiro

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    Think about what happens to the variances.
     
  6. Aug 31, 2012 #5
    It seems like bounded here means all most surely bounded. At least that's how Hoeffding inequality seems to be given elsewhere. I guess it then means that [itex]z=\bar{x}-\bar{y}[/itex] is bounded a.s. between [itex][\mu_x-\mu_y-\frac{1}{2}\sqrt{m^{-1}+n^{-1}}, \ \mu_x-\mu_y+\frac{1}{2}\sqrt{m^{-1}+n^{-1}}][/itex].?

    Thanks again for your help!
     
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