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\begin{align} P(h(\overline{X})-h(p)\geq \varepsilon)&\leq E[e^{-plog(\overline{X})-(1-p)log(1-\overline{X})}e^{plog(p)+(1-p)log(1-p)}]e^{-\varepsilon}\\ &=E[\frac{1}{\overline{X}^{p}(1-\overline{X})^{1-p}}]\frac{1}{p^{p}(1-p)^{1-p}}e^{-\varepsilon}\\ \end{align} So now we have to compute $$E[\frac{1}{(\sum X_{i})^{p}(n-\sum X_{i})^{1-p}}].$$ We ...

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No, it is not. You are maximizing the function on $[0,1]$, while you should maximize it on $[0,\frac{1}{2}]$. From the definition of MLE $\hat{p}(x)=\arg\max \{L(x,p):p \in[0,\frac{1}{2}] \}$. The correct answer is $\bar{p}=\min\{\bar{x} ,\frac{1}{2} \}$. Also note that, if initially $p \in (0,\frac{1}{2})$, then maximum likehood estimator wouldn't exist.

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Sorry, I eventually realize it is a simple question. Here is the answer of mine. first, for a semi-definite matrix $\Sigma$, we can find $\Lambda$ so that $$\Sigma = \Lambda \Lambda^T$$ Then, we have $$\Lambda^{-1} v ∼ \mathcal{N}(0,I)$$ So, we can apply the conventional LS to modified measurement process:  \Lambda^{-1}y ...

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Suppose you want to compute a realization at times $t_1<t_2<\dots<t_N$. This amounts to sampling from the distribution of an $N$ dimensional Gaussian vector $X$ with mean $0$ and covariance matrix $R$ where $R_{ij}=C(t_i-t_j)$. Such a vector is given by $X=R^{1/2} Y$, where $Y$ is an $N$ dimensional vector of independent standard normal variables ...

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