### 统计代写|似然估计作业代写Probability and Estimation代考|Bias and Weighted L2 Risks of Estimators

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## 统计代写|似然估计作业代写Probability and Estimation代考|Bias and Weighted L2 Risks of Estimators

This section contains the bias and the weighted $\mathrm{L}{2}$-risk expressions of the estimators. We study the comparative performance of the seven estimators defined on the basis of the weighted $L{2}$ risks defined by
\begin{aligned} R\left(\theta_{n}^{}: W_{1}, W_{2}\right)=& \mathbb{E}\left[\left(\theta_{1 n}^{}-\theta_{1}\right)^{\top} W_{1}\left(\theta_{1 n}^{}-\theta_{1}\right)\right] \ &+\mathbb{E}\left[\left(\theta_{2 n}^{}-\theta_{2}\right)^{\top} W_{2}\left(\theta_{2 n}^{}-\theta_{2}\right)\right] \end{aligned} where $\theta_{n}^{}=\left(\theta_{1 n}^{}, \theta_{2 n}^{}\right)^{\top}$ is any estimator of $\theta=\left(\theta_{1}^{\top}, \theta_{2}^{\top}\right)^{\top}$, and $W_{1}$ and $W_{2}$ are weight matrices. For convenience, when $W_{1}=I_{p_{1}}$ and $W_{2}=I_{p_{2}}$, we get the mean squared error (MSE) and write $R\left(\theta_{n}^{}: I_{p}\right)=\mathbb{E}\left[\left|\theta_{n}^{}-\theta\right|^{2}\right]$.
First, we note that for LSE,
\begin{aligned} b\left(\tilde{\theta}{n}\right) &=\mathbf{0} \ R\left(\tilde{\theta}{n}: N_{1}, N_{2}\right) &=\sigma^{2}\left(p_{1}+p_{2}\right) \end{aligned}
and for RLSE, $\hat{\theta}{\mathrm{R}}=\left(\tilde{\theta}{1 n}^{\top}, \mathbf{0}^{\top}\right)^{\top}$, we have
\begin{aligned} b\left(\hat{\theta}{\mathbb{R}}\right) &=\left(\mathbf{0}^{\top}, \boldsymbol{\theta}{2}^{\top}\right) \ R\left(\hat{\theta}{\mathbb{R}} ; N{1}, N_{2}\right) &=\sigma^{2}\left(p_{1}+\Delta^{2}\right) \end{aligned}

## 统计代写|似然估计作业代写Probability and Estimation代考|Hard Threshold Estimator

The bias of this estimator is given by
$$b\left(\hat{\theta}{n}^{\mathrm{HT}}(\kappa)\right)=\left(-\sigma n{j}^{-\frac{1}{2}} \Delta_{j} H_{3}\left(\kappa^{2} ; \Delta_{j}^{2}\right) \mid j=1, \ldots, p\right)^{\top}$$
where $H_{v}\left(; \Delta_{j}^{2}\right.$ ) is the cumulative distribution function (c.d.f.) of a noncentral $\chi^{2}$ distribution with $v$ DF. and noncentrality parameter $\Delta_{j}^{2}(j=1, \ldots, p)$.
The MSE of $\hat{\theta}{n}^{\mathrm{HT}}(\kappa)$ is given by \begin{aligned} R\left(\hat{\theta}{n}^{\mathrm{HT}}(\kappa): I_{p}\right)=& \sum_{j=1}^{p} \mathbb{E}\left[\tilde{\theta}{j n} I\left(\left|\tilde{\theta}{j n}\right|>\kappa \sigma n_{j}^{-\frac{1}{2}}\right)-\theta_{j}\right]^{2} \ =& \sigma^{2} \sum_{j=1}^{p} n_{j}^{-1}\left{\left(1-H_{3}\left(\kappa^{2} ; \Delta_{j}^{2}\right)\right)\right.\ &\left.+\Delta_{j}^{2}\left(2 H_{3}\left(\kappa^{2} ; \Delta_{j}^{2}\right)-H_{5}\left(\kappa^{2} ; \Delta_{j}^{2}\right)\right)\right} \end{aligned}

Since $\left[\tilde{\theta}{j m} I\left(\left|\tilde{\theta}{j n}\right|>\kappa \sigma n_{j}^{-\frac{1}{2}}\right)-\theta_{j}\right]^{2} \leq\left(\tilde{\theta}{j n}-\theta{j}\right)^{2}+\theta_{j}^{2}$, we obtain $R\left(\hat{\theta}{n}^{\mathrm{HT}}(\kappa): I{p}\right) \leq \sigma^{2} \operatorname{tr} N^{-1}+\theta^{\top} \boldsymbol{\theta} \quad$ (free of $\left.\kappa\right) .$
Following Donoho and Johnstone (1994), one can show that what follows holds:
where $\rho_{\mathrm{HT}}(\kappa, 0)=2[(1-\Phi(\kappa))+\kappa \varphi(\kappa)]$, and $\varphi(\cdot)$ and $\Phi(\cdot)$ are the probability density function (p.d.f.) and c.d.f. of standard normal distribution, respectively.
Theorem 3.1 Under the assumed regularity conditions, the weighted $\mathrm{L}{2}$-risk bounds are given by $$R\left(\hat{\boldsymbol{\theta}}{n}^{\mathrm{HT}}(\kappa): N_{1}, N_{2}\right) \leq\left{\begin{array}{lll} \text { (i) } & \sigma^{2}\left(1+\kappa^{2}\right)\left(p_{1}+p_{2}\right) & \kappa>1, \ \text { (ii) } & \sigma^{2}\left(p_{1}+p_{2}\right)+\boldsymbol{\theta}{1}^{\top} N{1} \boldsymbol{\theta}{1} & \ & +\boldsymbol{\theta}{2}^{\top} \boldsymbol{N}{2} \boldsymbol{\theta}{2} & \forall \boldsymbol{\theta} \in \mathbb{R}^{p}, \ \text { (iii) } & \sigma^{2} \rho_{\mathrm{HT}}(\kappa, 0)\left(p_{1}+p_{2}\right) & \ & +1.2\left{\boldsymbol{\theta}{1}^{\top} \boldsymbol{N}{1} \boldsymbol{\theta}{1}+\boldsymbol{\theta}{2}^{\top} N_{2} \boldsymbol{\theta}{2}\right} & 0<\boldsymbol{\theta}{p}^{\top} \end{array}\right.$$
If the solution of $\hat{\theta}{n}^{\mathrm{HT}}(\kappa)$ has the configuration $\left(\tilde{\theta}{1 n}^{\mathrm{T}}, \mathbf{0}^{\mathrm{T}}\right)^{\mathrm{T}}$, then the $\mathrm{L}{2}$ risk of $\hat{\boldsymbol{\theta}}{n}^{\mathrm{HT}}(\kappa)$ is given by
$$R\left(\hat{\boldsymbol{\theta}}{n}^{\mathrm{HT}}(\kappa): \boldsymbol{N}{1}, \boldsymbol{N}{2}\right)=\sigma^{2}\left[p{1}+\Delta^{2}\right],$$
independent of $\kappa$.

## 统计代写|似然估计作业代写Probability and Estimation代考|LASSO Estimator

The bias expression of the LASSO estimator is given by
$$b\left(\theta_{n}^{\mathrm{L}}(\lambda)\right)=\left(\sigma n_{j}^{-\frac{1}{2}}\left[\lambda\left(2 \Phi\left(\Delta_{j}\right)-1\right) ; j=1, \ldots, p_{1} ;-\Delta_{p_{1}+1}, \ldots, \Delta_{p}\right)^{\top} .\right.$$
The MSE of the LASSO estimator has the form
$$R\left(\hat{\boldsymbol{\theta}}{n}^{\mathrm{L}}(\lambda): \boldsymbol{I}{p}\right)=\sigma^{2} \sum_{j=1}^{p_{1}} n_{j}^{-1} \rho_{\mathrm{ST}}\left(\lambda, \Delta_{j}\right)+\Delta^{2},$$

where
\begin{aligned} \rho_{\mathrm{ST}}\left(\lambda, \Delta_{j}\right)=&\left(1+\lambda^{2}\right)\left{1-\Phi\left(\lambda-\Delta_{j}\right)+\Phi\left(-\lambda-\Delta_{j}\right)\right} \ &+\Delta_{j}^{2}\left{\Phi\left(\lambda-\Delta_{j}\right)-\Phi\left(-\lambda-\Delta_{j}\right)\right} \ &-\left{\left(\lambda-\Delta_{j}\right) \varphi\left(\lambda+\Delta_{j}\right)+\left(\lambda+\Delta_{j}\right) \varphi\left(\lambda-\Delta_{j}\right)\right} \end{aligned}
Thus, according to Donoho and Johnstone (1994, Appendix 2), we have the following result.
Under the assumed regularity conditions,
$R\left(\hat{\theta}{n}^{\mathrm{L}}(\lambda): \boldsymbol{I}{p}\right) \leq\left{\begin{array}{lll}\text { (i) } & \sigma^{2}\left(1+\lambda^{2}\right) \operatorname{tr} N^{-1} & \forall \theta \in \mathbb{R}^{p}, \kappa \ \text { (ii) } & \sigma^{2} \operatorname{tr} N^{-1}+\theta^{\top} \boldsymbol{\theta} & \forall \theta \in \mathbb{R}^{p}, \ \text { (iii) } & \sigma^{2} \rho_{\mathrm{ST}}(\lambda, 0) \operatorname{tr} \boldsymbol{N}^{-1}+1.2 \boldsymbol{\theta}^{\top} \boldsymbol{\theta} & \forall \theta \in \mathbb{R}^{p},\end{array}\right.$
where $\rho_{\mathrm{ST}}(\lambda, 0)=2\left[\left(1+\lambda^{2}\right)(1-\Phi(\lambda))-\kappa \phi(\lambda)\right]$.
If the solution of $\hat{\theta}{n}^{L}(\lambda)$ has the configuration $\left(\hat{\theta}{1 n}^{\top}, \mathbf{0}^{\top}\right)$, then the $L_{2}$ risk of $\hat{\theta}{n}^{\mathrm{L}}(\lambda)$ is given by $$R\left(\hat{\boldsymbol{\theta}}{n}^{\mathrm{L}}(\lambda): N_{1}, N_{2}\right)=\sigma^{2}\left(p_{1}+\Delta^{2}\right)$$
Thus, we note that
$$R\left(\hat{\boldsymbol{\theta}}{n} ; \boldsymbol{N}{1}, \boldsymbol{N}{2}\right)=R\left(\hat{\boldsymbol{\theta}}{n}^{\mathrm{HT}}(\kappa) ; \boldsymbol{N}{1}, \boldsymbol{N}{2}\right)=R\left(\hat{\boldsymbol{\theta}}{n}^{\mathrm{L}}(\lambda) ; \boldsymbol{N}{1}, \boldsymbol{N}{2}\right)=\sigma^{2}\left(p{1}+\Delta^{2}\right) .$$
To prove the $\mathrm{L}{2}$ risk of LASSO, we consider the multivariate decision theory. We are given the LSE of $\theta$ as $\tilde{\theta}{n}=\left(\tilde{\theta}{1 n}, \ldots, \tilde{\theta}{p n}\right)^{\top}$ according to
$$\tilde{\theta}{j n}=\theta{j}+\sigma n_{j}^{-\frac{1}{2}} \mathcal{Z}{j}, \quad \mathcal{Z}{j} \sim \mathcal{N}(0,1),$$
where $\sigma n_{j}^{-\frac{1}{2}}$ is the marginal variance of $\tilde{\theta}{j n}$ and noise level, and $\left{\theta{j}\right}_{j=1, \ldots, p}$ are the treatment effects of interest. We measure the quality of the estimators based on the $\mathrm{L}{2}$ risk, $R\left(\tilde{\boldsymbol{\theta}}{n}: \boldsymbol{I}{p}\right)=\mathbb{E}\left[\left|\tilde{\boldsymbol{\theta}}{n}-\boldsymbol{\theta}\right|^{2}\right]$. Note that for a sparse solution, we use (3.11).
Consider the family of diagonal linear projections,
$$T_{\mathrm{DP}}\left(\hat{\theta}{n}^{\mathrm{L}}(\lambda): \delta\right)=\left(\delta{1} \hat{\theta}{1 n}^{\mathrm{L}}(\lambda), \ldots, \delta{p} \hat{\theta}{p n}^{\mathrm{L}}(\lambda)\right)^{\top},$$ where $\delta=\left(\delta{1}, \ldots, \delta_{p}\right)^{\top}, \delta_{j} \in(0,1), j=1, \ldots, p$. Such estimators “kill” or “keep” the coordinates.

Suppose we had available an oracle which would supply for us the coefficients $\delta_{j}$ optimal for use in the diagonal projection scheme (3.15). These “ideal” coefficients are $\delta_{j}=I\left(\left|\theta_{j}\right|>\sigma n_{j}^{-\frac{1}{2}}\right)$. Ideal diagonal projections consist of estimating only those $\theta_{j}$, which are larger than its noise, $\sigma n_{j}^{-\frac{1}{2}}(j=1, \ldots, p)$. These yield the “ideal” $\mathrm{L}_{2}$ risk given by (3.16).

## 统计代写|似然估计作业代写Probability and Estimation代考|Bias and Weighted L2 Risks of Estimators

R(θn:在1,在2)=和[(θ1n−θ1)⊤在1(θ1n−θ1)] +和[(θ2n−θ2)⊤在2(θ2n−θ2)]在哪里θn=(θ1n,θ2n)⊤是任何估计量θ=(θ1⊤,θ2⊤)⊤， 和在1和在2是权重矩阵。为方便起见，当在1=一世p1和在2=一世p2，我们得到均方误差（MSE）并写R(θn:一世p)=和[|θn−θ|2].

b(θ~n)=0 R(θ~n:ñ1,ñ2)=σ2(p1+p2)

b(θ^R)=(0⊤,θ2⊤) R(θ^R;ñ1,ñ2)=σ2(p1+Δ2)

## 统计代写|似然估计作业代写Probability and Estimation代考|Hard Threshold Estimator

b(θ^nH吨(ķ))=(−σnj−12ΔjH3(ķ2;Δj2)∣j=1,…,p)⊤

## 统计代写|似然估计作业代写Probability and Estimation代考|LASSO Estimator

LASSO 估计器的偏差表达式由下式给出
b(θn大号(λ))=(σnj−12[λ(2披(Δj)−1);j=1,…,p1;−Δp1+1,…,Δp)⊤.
LASSO 估计器的 MSE 具有以下形式
R(θ^n大号(λ):一世p)=σ2∑j=1p1nj−1ρ小号吨(λ,Δj)+Δ2,

\begin{对齐} \rho_{\mathrm{ST}}\left(\lambda, \Delta_{j}\right)=&\left(1+\lambda^{2}\right)\left{1-\ Phi\left(\lambda-\Delta_{j}\right)+\Phi\left(-\lambda-\Delta_{j}\right)\right} \ &+\Delta_{j}^{2}\left {\Phi\left(\lambda-\Delta_{j}\right)-\Phi\left(-\lambda-\Delta_{j}\right)\right} \ &-\left{\left(\lambda- \Delta_{j}\right) \varphi\left(\lambda+\Delta_{j}\right)+\left(\lambda+\Delta_{j}\right) \varphi\left(\lambda-\Delta_{j} \right)\right} \end{对齐}\begin{对齐} \rho_{\mathrm{ST}}\left(\lambda, \Delta_{j}\right)=&\left(1+\lambda^{2}\right)\left{1-\ Phi\left(\lambda-\Delta_{j}\right)+\Phi\left(-\lambda-\Delta_{j}\right)\right} \ &+\Delta_{j}^{2}\left {\Phi\left(\lambda-\Delta_{j}\right)-\Phi\left(-\lambda-\Delta_{j}\right)\right} \ &-\left{\left(\lambda- \Delta_{j}\right) \varphi\left(\lambda+\Delta_{j}\right)+\left(\lambda+\Delta_{j}\right) \varphi\left(\lambda-\Delta_{j} \right)\right} \end{对齐}

\$R\left(\hat{\theta}{n}^{\mathrm{L}}(\lambda): \boldsymbol{I}{p}\right) \leq\left{ （一世） σ2(1+λ2)tr⁡ñ−1∀θ∈Rp,ķ  (二) σ2tr⁡ñ−1+θ⊤θ∀θ∈Rp,  ㈢ σ2ρ小号吨(λ,0)tr⁡ñ−1+1.2θ⊤θ∀θ∈Rp,\对。在H和r和\rho_{\mathrm{ST}}(\lambda, 0)=2\left[\left(1+\lambda^{2}\right)(1-\Phi(\lambda))-\kappa \phi( \lambda)\右].一世F吨H和s这l在吨一世这n这F\hat{\theta}{n}^{L}(\lambda)H一种s吨H和C这nF一世G在r一种吨一世这n\left(\hat{\theta}{1 n}^{\top}, \mathbf{0}^{\top}\right),吨H和n吨H和L_{2}r一世sķ这F\hat {\theta {n} ^ {\ mathrm {L}} (\lambda)一世sG一世在和nb是R(θ^n大号(λ):ñ1,ñ2)=σ2(p1+Δ2)吨H在s,在和n这吨和吨H一种吨R(θ^n;ñ1,ñ2)=R(θ^nH吨(ķ);ñ1,ñ2)=R(θ^n大号(λ);ñ1,ñ2)=σ2(p1+Δ2).吨这pr这在和吨H和\数学{L} {2r一世sķ这F大号一种小号小号这,在和C这ns一世d和r吨H和米在l吨一世在一种r一世一种吨和d和C一世s一世这n吨H和这r是.在和一种r和G一世在和n吨H和大号小号和这F\θ一种s\tilde{\theta}{n}=\left(\tilde{\theta}{1 n}, \ldots, \tilde{\theta}{pn}\right)^{\top}一种CC这rd一世nG吨这θ~jn=θj+σnj−12从j,从j∼ñ(0,1),在H和r和\sigma n_{j}^{-\frac{1}{2}}一世s吨H和米一种rG一世n一种l在一种r一世一种nC和这F\波浪号{\theta}{jn}一种ndn这一世s和l和在和l,一种nd\left{\theta{j}\right}_{j=1, \ldots, p}一种r和吨H和吨r和一种吨米和n吨和FF和C吨s这F一世n吨和r和s吨.在和米和一种s在r和吨H和q在一种l一世吨是这F吨H和和s吨一世米一种吨这rsb一种s和d这n吨H和\数学{L} {2r一世sķ,R\left(\tilde{\boldsymbol{\theta}}{n}: \boldsymbol{I}{p}\right)=\mathbb{E}\left[\left|\tilde{\boldsymbol{\theta} }{n}-\boldsymbol{\theta}\right|^{2}\right].ñ这吨和吨H一种吨F这r一种sp一种rs和s这l在吨一世这n,在和在s和(3.11).C这ns一世d和r吨H和F一种米一世l是这Fd一世一种G这n一种ll一世n和一种rpr这j和C吨一世这ns,吨D磷(θ^n大号(λ):d)=(d1θ^1n大号(λ),…,dpθ^pn大号(λ))⊤,在H和r和\delta=\left(\delta{1}, \ldots, \delta_{p}\right)^{\top}, \delta_{j} \in(0,1), j=1, \ldots, p美元。这样的估计器“杀死”或“保留”坐标。

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