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Suppose iid $X_1,\ldots, X_n\sim p(x;\theta)$ and suppose $T_n(X_1,\ldots, X_n)$ has pdf $$ p_{T_n(X)}(x) = \frac{1}{n}\sum_{i=1}^n \log \frac{p(x_i;\theta_1)}{p(x_i,\theta_2)} $$ where $\theta_1$ and $\theta_2$ are two true value of $\theta$. I wish to prove that $\sqrt{n}(T_n-\mu)\to N(0,\sigma)$ for some $\sigma$, where $\mu$ stands for the mean of $T_n$.

My try: I plan to use central limit theorem (CLT). I thought, in the beginning, this question is only a trivial application of $CLT$ but now I am a bit confused. Indeed, if I want to use CLT, I am actually assuming that the r.v $Y$ with pdf $p_Y(y) = \log\frac{p(y;\theta_1)}{p(y;\theta_2)}$ is iid. Is it true? and if yes, what is the mean value of $T(X)$ and what is the value of $\sigma$?

Thank you!

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  • $\begingroup$ There is no reason to think that $T_n$ is an average of iid variables in this context. But it might help to notice that the sum of logs is the log of the product... $\endgroup$ – Ian Oct 7 '15 at 18:47
  • $\begingroup$ @Ian I noticed. Actually the problem was originally give the $T_n$ in the product form. I changed it to summation form so that I can use CLT. But it seems wrong now... $\endgroup$ – spatially Oct 7 '15 at 18:55
  • $\begingroup$ No reason to expect this to be a PDF. $\endgroup$ – Did Oct 7 '15 at 20:26
  • $\begingroup$ @Did hmm, you are right. But I think it is still possible to show what I want by using CLT... $\endgroup$ – spatially Oct 7 '15 at 20:55
  • $\begingroup$ You need to have something random in the right hand side of your equation. Perhaps $X_{i}$ in for $x_{i}$? In this case you would have $\overline{Y}$ where $Y_{i}=p(X_{i};\theta_{1})/p(X_{i};\theta_{2})$. $\endgroup$ – Rubarb Oct 9 '15 at 1:17

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