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3

The answer is no. But your continuity requirement makes it more difficult to find an explicit (constructed) example. Define the function $\varphi$ on $\mathbb{R}$ as: $$\forall x\in\mathbb{R},\quad\varphi(x)=\begin{cases}\sin(x)&\text{if x\in[0,\pi]}\\0&\text{otherwise.}\end{cases}$$ Clearly, the function $\varphi$ is continuous on $\mathbb{R}$ ...

1

As written by Martín-Blas Pérez Pinilla, let us suppose that you want to find the optimum values of parameters $A$ and $B$ which minimize the objective function $$\Phi(A,B)=\sum _{n=1}^N (\alpha_n-A\cos (Bt_n))^2+(\beta_n+A\sin (Bt_n))^2=\sum _{n=1}^N r_n$$ Now, since you want the objective function to be minimum, write its derivatives with respect to $A$ ...

0

$$f(t)=Ae^{−iBt}=A(\cos(Bt)-i\sin(Bt))$$ $$\alpha_n+\beta_n i=f(t_n)=A(\cos(Bt_n)-i\sin(Bt_n))$$ $$A=|\alpha_n+\beta_n i|=\sqrt{\alpha_n^2+\beta_n^2}$$ $$B= -\frac1{t_n}\arctan\frac{\alpha_n}{\beta_n}$$ $$\cdots$$

0

I assume from your question that $s_{MS} = \mathbb{E}(s | r)$. I will also assume that $h$ is independent of all other variables. First note that $\mathbb{E}(s|r)$ is a marginal distribution over $h$. Thus, let's deploy iterated expectation: \begin{align} \mathbb{E}(s|r) &= \mathbb{E}[\mathbb{E}(s|r, h)]\\ &= ... 1 You seem to be mixing together various incompatible notations and various incompatible objects quite a lot... For example, the expectation of the estimator is not what you write (which is absurd) but $$E_\theta(\beta(X))=\int_{[0,\theta]^n}\max\{x_1,x_2,\ldots,x_n\}\,\theta^{-n}\mathrm dx_1\mathrm dx_2\ldots\mathrm dx_n.$$ Alternatively, if one knows ... 1 For every random variableY\$, one has $$c^2\cdot\mathbf 1_{|Y|\geqslant c}\leqslant Y^2.$$ Integrating, one gets $$c^2\cdot P(|Y|\geqslant c)\leqslant E(Y^2).$$ Apply this to $$Y=\text{____}.$$ And the thing even has a name...

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