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Unless you are a mathematician interested in the calculus of variations approach, before going into optimal control and estimation I'd start with a more introductory and less specialized text on classical control such as Feedback Systems An Introduction for Scientists and Engineers Karl Johan Åström Richard M. Murray There are many other good textbooks for ...

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Ok i found the paper that you were working. The question is actually easier, (question 8b). You have $n$ samples for the left eye and $n$ samples for the right eye and they are all independent and identicallly distributed. You should find the maximum estimator for a $2n$ sample (define $X=(L_1,\cdots,L_n,R_1,\cdots,R_n)$ as your sample). Is the classical ...

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Let $L_1,\cdots,L_n$ follow Normal$(\mu_1,\sigma^{2}_{1})$ and $R_1,\cdots,R_n$ follow Normal$(\mu_2,\sigma^{2}_{2})$. Let also $\mathbf{X^{T}_{i}}=(L_i,R_i)$ and $\boldsymbol{\mu}=(\mu_1,\mu_2)'\ , \boldsymbol{\Sigma}=cov(\mathbf{X})$. Then their likelihood function is ...

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$(n-1)\frac{\hat \sigma^2}{\sigma^2}$ will be Chi-square distributed with n-1 degrees of freedom. So the SE will be the standard deviation of the associated Chi-squared distribution $\sqrt{2(n-1)}$ multiplied by $(\frac{\sigma^2}{n-1})^{1/2}$. Unfortunately, you will not know $\sigma^2$,so you will not know the standard error of the sample variance. However, ...

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Everything looks right except that at step three it should be $1+\theta$ instead of $1-\theta$.

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