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You may also want to look at the paper linked here: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1937879 It details how to consistently estimate the signal and noise variances (as well as measurement parameters, if you have those) using an EM algorithm. The application may not be generalizable to yours, though, so beware.

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The first definition is the usual one; see e.g. Wikipedia. Regarding your question in a comment whether the second expression is used, since this is the expected bias, you could search for "expected bias" or "expected value of the bias" to find examples of its use.

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I don't think I have enough of your problem to answer that question, but if you look at Huber's Robust Statistics or Keener's Theoretical Statistics, the sections on M-estimators might help. For example, in Keener, the maximum likelihood estimator is characterized as when the derivative of the likelihood equals 0. They are able to do proofs using a Taylor ...

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For $E[(X-E[X])^2]$, the minimality of $c=E[X]$ can be understood from a number of vantage points. On the one hand, it is directly related to the abstract definition of conditional expectation, which asks to minimize $E[(X-Y)^2|F]$ where $F$ is the sigma algebra of constant functions. On the other hand, $$E[(X-c)^2]=E[(X-E[X]+E[X]-c)^2]=E[(X-E[X])^2]+(E[X]-... 1 Perhaps I misunderstand the setup, but isn't \tau the mean of the distribution?$$\tau = \int x \mathop{dF(x)} =\frac{1}{b-a} \int_a^b x \mathop{dx} = \frac{1}{b-a} \frac{1}{2} [x^2]_{x=a}^b = \frac{1}{2} \frac{b^2-a^2}{b-a} = \frac{a+b}{2}. Your log likelihood computation is almost correct. \begin{align} L(X^n,a,b) = (b-a)^{-n} \cdot \mathbf{1}[\...

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The statistic $T_2$ does not have a $t_n$ distribution. The issue is that the numerator $\overline{X_n}-\mu_0$ is not independent of the denominator $\sqrt{R}$, unlike the situation with $T_1$. More importantly, using $T_2$ will not lead to a more powerful test; the Neyman-Pearson lemma can be used to show that $T_1$ provides a uniformly most powerful test, ...

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