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Uniform Convergence problem

  1. Sep 27, 2008 #1
    First of all, hello all this is my first post i think. Congratulations on this great community
    Please move my post if I'm not posting on the right forum and I'm sorry for any inconvenience.

    I have this problem that I need to solve and I don't have a clue. I hope you could give me some ideas.

    I need to show this: [tex]\mathop {\sup }\limits_\theta \left\| {E_\theta \left( {\hat \beta _T } \right) - b\left( \theta \right)} \right\| \to 0[/tex]

    knowing that:
    [tex]\hat \beta _T \stackrel{T\rightarrow\infty}{\rightarrow} b\left( {\theta _0 } \right)[/tex]

    where [tex]\hat \beta _T[/tex] is a stochastic function of [tex]y_T[/tex] that comes from a distribution with true parameter [tex]\theta_0[/tex]

    θ and β belongs in a compact subset of R^p and R^q respectively.

    The convergence apparently is non-stochastic as we've taken expectation.
    A hint is to add and subtract something into the norm and use the triangle inequality to show the above claim. But, I have no idea how to treat the expectation.

    I haven't supplied all info there is, but please tell me if you can think of any possible approaches for this. Any help is much appreciated
    Last edited: Sep 27, 2008
  2. jcsd
  3. Sep 28, 2008 #2
    I get the feeling you should add and subtract [tex] \hat \beta_T[/tex] and use its convergence and the strong law of large numbers maybe.
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