### 统计代写|似然估计作业代写Probability and Estimation代考|Comparison of Bias and MSE Functions

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## 统计代写|似然估计作业代写Probability and Estimation代考|Comparison of Bias and MSE Functions

Since the bias and MSE expressions are known to us, we may compare them for the three estimators, namely, $\tilde{\theta}{n}, \hat{\theta}{n}$, and $\hat{\theta}{n}^{\mathrm{PT}}(\alpha)$ as well as $\tilde{\beta}{n}, \hat{\beta}{n}$, and $\hat{\beta}{n}^{\mathrm{PT}}(\alpha)$. Note that all the expressions are functions of $\Delta^{2}$, which is the noncentrality parameter of the noncentral $F$-distribution. Also, $\Delta^{2}$ is the standardized distance between $\beta$ and $\beta_{\mathrm{o}}$. First, we compare the bias functions as in Theorem $2.4$, when $\sigma^{2}$ is unknown.
For $\bar{x}=0$ or under $\mathcal{H}{\mathrm{o}}$, \begin{aligned} &b\left(\tilde{\theta}{n}\right)=b\left(\hat{\theta}{n}\right)=b\left(\hat{\theta}{n}^{\mathrm{pT}}(\alpha)\right)=0 \ &b\left(\tilde{\beta}{n}\right)=b\left(\hat{\beta}{n}\right)=b\left(\hat{\beta}_{n}^{\mathrm{PT}}(\alpha)\right)=0 . \end{aligned}

Otherwise, for all $\Delta^{2}$ and $\bar{x} \neq 0$,
\begin{aligned} &0=b\left(\tilde{\theta}{n}\right) \leq\left|b\left(\hat{\theta}{n}^{\mathrm{pT}}(\alpha)\right)\right|=\left|\beta-\beta_{\mathrm{o}}\right| \bar{x}{3, m}\left(\frac{1}{3} F{1, m}(\alpha) ; \Delta^{2}\right) \leq\left|b\left(\hat{\theta}{n}\right)\right| \ &0=b\left(\tilde{\beta}{n}\right) \leq\left|b\left(\hat{\beta}{n}^{\mathrm{pT}}(\alpha)\right)\right|=\left|\beta-\beta{\mathrm{o}}\right| G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; \Delta^{2}\right) \leq\left|b\left(\hat{\beta}{n}\right)\right| . \end{aligned} The absolute bias of $\hat{\theta}{n}$ is linear in $\Delta^{2}$, while the absolute bias of $\hat{\theta}{n}^{\mathrm{pT}}(\alpha)$ increases to the maximum as $\Delta^{2}$ moves away from the origin, and then decreases toward zero as $\Delta^{2} \rightarrow \infty$. Similar conclusions hold for $\hat{\beta}{n}^{\mathrm{PT}}(\alpha)$.

Now, we compare the MSE functions of the restricted estimators and PTEs with respect to the traditional estimator, $\tilde{\theta}{n}$ and $\tilde{\beta}{n}$, respectively. The REff of $\hat{\theta}{n}$ compared to $\tilde{\theta}{n}$ may be written as
$$\operatorname{REff}\left(\hat{\theta}{n}: \tilde{\theta}{n}\right)=\left(1+\frac{n \bar{x}^{2}}{Q}\right)\left[1+\frac{n \bar{x}^{2}}{Q} \Delta^{2}\right]^{-1} \text {. }$$
The efficiency is a decreasing function of $\Delta^{2}$. Under $\mathcal{H}{\mathrm{o}}$ (i.e. $\Delta^{2}=0$ ), it has the maximum value $$\operatorname{REff}\left(\hat{\theta}{n} ; \tilde{\theta}{n}\right)=\left(1+\frac{n \bar{x}^{2}}{Q}\right) \geq 1,$$ and $\operatorname{REff}\left(\hat{\theta}{n} ; \tilde{\theta}{n}\right) \geq 1$, accordingly, as $\Delta^{2} \geq 1$. Thus, $\hat{\theta}{n}$ performs better than $\tilde{\theta}{n}$ whenever $\Delta^{2}<1$; otherwise, $\tilde{\theta}{n}$ performs better $\hat{\theta}{n}$. The REff of $\hat{\theta}{n}^{\mathrm{pT}}(\alpha)$ compared to $\tilde{\theta}{n}$ may be written as $$\operatorname{REff}\left(\hat{\theta}{n}^{\mathrm{PT}}(\alpha) ; \tilde{\theta}{n}\right)=\left[1+g\left(\Delta^{2}\right)\right]^{-1},$$ where \begin{aligned} g\left(\Delta^{2}\right)=&-\frac{\bar{x}^{2}}{Q}\left(\frac{1}{n}+\frac{\bar{x}^{2}}{Q}\right)^{-1}\left[G{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; \Delta^{2}\right)\right.\ &\left.-\Delta^{2}\left{2 G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; \Delta^{2}\right)-G_{5, m}\left(\frac{1}{5} F_{1, m}(\alpha) ; \Delta^{2}\right)\right}\right] \end{aligned}
Under the $\mathcal{H}{0}$, it has the maximum value $$\operatorname{REff}\left(\hat{\theta}{n}^{\mathrm{pT}}(\alpha) ; \tilde{\theta}{n}\right)=\left{1-\frac{\bar{x}^{2}}{Q}\left(\frac{1}{n}+\frac{\bar{x}^{2}}{Q}\right)^{-1} G{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; 0\right)\right}^{-1} \geq 1$$
and $\operatorname{REff}\left(\hat{\theta}{n}^{\mathrm{pT}}(\alpha) ; \ddot{\theta}{n}\right)$ according as
$$\Delta^{2} \leq \Delta^{2}(\alpha)=\frac{G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; \Delta^{2}\right)}{2 G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; \Delta^{2}\right)-G_{5, m}\left(\frac{1}{5} F_{1, m}(\alpha) ; \Delta^{2}\right)}$$

## 统计代写|似然估计作业代写Probability and Estimation代考|Alternative PTE

In this subsection, we provide the alternative expressions for the estimator of PT and its bias and MSE. To test the hypothesis $\mathcal{H}{0}: \beta=0$ vs. $\mathcal{H}{\mathrm{A}}: \beta \neq 0$, we use the following test statistic:
$$Z_{n}=\frac{\sqrt{Q} \tilde{\beta}{n}}{\sigma} .$$ The PTE of $\beta$ is given by \begin{aligned} \hat{\beta}{n}^{\mathrm{PT}}(\alpha) &=\tilde{\beta}{n}-\tilde{\beta}{n} I\left(\left|\tilde{\beta}{n}\right|<\frac{\lambda \sigma}{\sqrt{Q}}\right) \ &=\frac{\sigma}{\sqrt{Q}}\left[Z{n}-Z_{n} I\left(\left|Z_{n}\right|<\lambda\right)\right] \end{aligned}
where $\lambda=\sqrt{2 \log 2}$.
Hence, the bias of $\tilde{\beta}{n}$ equals $\beta[\Phi(\lambda-\Delta)-\Phi(-\lambda-\Delta)]-[\phi(\lambda-\Delta)-\phi(\lambda+$ $\Delta)$, and the MSE is given by $$\operatorname{MSE}\left(\hat{\beta}{n}^{\mathrm{PT}}\right)=\frac{\sigma^{2}}{Q} \rho_{\mathrm{PT}}(\lambda, \Delta)$$

## 统计代写|似然估计作业代写Probability and Estimation代考|Optimum Level of Significance of Preliminary Test

Consider the REff of $\hat{\theta}{n}^{\mathrm{pT}}(\alpha)$ compared to $\tilde{\theta}{n^{*}}$ Denoting it by REff $\left(\alpha ; \Delta^{2}\right)$, we have
$$\operatorname{REff}\left(\alpha, \Delta^{2}\right)=\left[1+g\left(\Delta^{2}\right)\right]^{-1},$$
where
\begin{aligned} g\left(\Delta^{2}\right)=&-\frac{\bar{x}^{2}}{Q}\left(\frac{1}{n}+\frac{\bar{x}^{2}}{Q}\right)^{-1}\left[G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; \Delta^{2}\right)\right.\ &\left.-\Delta^{2}\left{2 G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; \Delta^{2}\right)-G_{5, m}\left(\frac{1}{5} F_{1, m}(\alpha) ; \Delta^{2}\right)\right}\right] \end{aligned}
The graph of REff $\left(\alpha, \Delta^{2}\right)$, as a function of $\Delta^{2}$ for fixed $\alpha$, is decreasing crossing the 1-line to a minimum at $\Delta^{2}=\Delta_{0}^{2}(\alpha)$ (say); then it increases toward the 1-line as $\Delta^{2} \rightarrow \infty$. The maximum value of $\operatorname{REff}\left(\alpha, \Delta^{2}\right)$ occurs at $\Delta^{2}=0$ with the value
$$\operatorname{REff}(\alpha ; 0)=\left{1-\frac{\bar{x}^{2}}{Q}\left(\frac{1}{n}+\frac{\bar{x}^{2}}{Q}\right)^{-1} G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; 0\right)\right}^{-1} \geq 1,$$
for all $\alpha \in A$, the set of possible values of $\alpha$. The value of REff $(\alpha ; 0)$ decreases as $\alpha$-values increase. On the other hand, if $\alpha=0$ and $\Delta^{2}$ vary, the graphs of $\operatorname{REff}\left(0, \Delta^{2}\right)$ and $\operatorname{REff}\left(1, \Delta^{2}\right)$ intersect at $\Delta^{2}=1$. In general, $\operatorname{REff}\left(\alpha_{1}, \Delta^{2}\right)$ and $\operatorname{REff}\left(\alpha_{2}, \Delta^{2}\right)$ intersect within the interval $0 \leq \Delta^{2} \leq 1$; the value of $\Delta^{2}$ at the intersection increases as $\alpha$-values increase. Therefore, for two different $\alpha$-values, $\operatorname{REff}\left(\alpha_{1}, \Delta^{2}\right)$ and $\operatorname{REff}\left(\alpha_{2}, \Delta^{2}\right)$ will always intersect below the 1 -line.
In order to obtain a PTE with a minimum guaranteed efficiency, $E_{0}$, we adopt the following procedure: If $0 \leq \Delta^{2} \leq 1$, we always choose $\tilde{\theta}{n}$, since $\operatorname{REff}\left(\alpha, \Delta^{2}\right) \geq 1$ in this interval. However, since in general $\Delta^{2}$ is unknown, there is no way to choose an estimate that is uniformly best. For this reason, we select an estimator with minimum guaranteed efficiency, such as $E{0}$, and look for a suitable $\alpha$ from the set, $A=\left{\alpha \mid \operatorname{REff}\left(\alpha, \Delta^{2}\right) \geq E_{0}\right}$. The estimator chosen

maximizes $\operatorname{REff}\left(\alpha, \Delta^{2}\right)$ over all $\alpha \in A$ and $\Delta^{2}$. Thus, we solve the following equation for the optimum $\alpha^{}$ : $$\min {\Delta^{2}} \operatorname{REff}\left(\alpha, \Delta^{2}\right)=E\left(\alpha, \Delta{0}^{2}(\alpha)\right)=E_{0} .$$
The solution $\alpha^{}$ obtained this way gives the PTE with minimum guaranteed efficiency $E_{0}$, which may increase toward $\operatorname{REff}\left(\alpha^{*}, 0\right)$ given by $(2.61)$, and Table $2.2$. For the following given data, we have computed the maximum and minimum guaranteed REff for the estimators of $\theta$ and provided them in Table 2.2.
\begin{aligned} x=&(19.383,21.117,18.99,19.415,20.394,20.212,20.163,20.521,20.125,\ & 19.944,18.345,21.45,19.479,20.199,20.677,19.661,20.114,19.724 \ &18.225,20.669)^{\top} \end{aligned}

## 统计代写|似然估计作业代写Probability and Estimation代考|Comparison of Bias and MSE Functions

0=b(θ~n)≤|b(θ^np吨(一种))|=|b−b这|X¯3,米(13F1,米(一种);Δ2)≤|b(θ^n)| 0=b(b~n)≤|b(b^np吨(一种))|=|b−b这|G3,米(13F1,米(一种);Δ2)≤|b(b^n)|.绝对偏差θ^n是线性的Δ2，而绝对偏差θ^np吨(一种)增加到最大值Δ2远离原点，然后随着Δ2→∞. 类似的结论适用于b^n磷吨(一种).

REff⁡(θ^n:θ~n)=(1+nX¯2问)[1+nX¯2问Δ2]−1.

Δ2≤Δ2(一种)=G3,米(13F1,米(一种);Δ2)2G3,米(13F1,米(一种);Δ2)−G5,米(15F1,米(一种);Δ2)

## 统计代写|似然估计作业代写Probability and Estimation代考|Optimum Level of Significance of Preliminary Test

REff⁡(一种,Δ2)=[1+G(Δ2)]−1,

\begin{aligned} g\left(\Delta^{2}\right)=&-\frac{\bar{x}^{2}}{Q}\left(\frac{1}{n}+\ frac{\bar{x}^{2}}{Q}\right)^{-1}\left[G_{3, m}\left(\frac{1}{3} F_{1, m}( \alpha) ; \Delta^{2}\right)\right.\ &\left.-\Delta^{2}\left{2 G_{3, m}\left(\frac{1}{3} F_ {1, m}(\alpha) ; \Delta^{2}\right)-G_{5, m}\left(\frac{1}{5} F_{1, m}(\alpha) ; \Delta ^{2}\right)\right}\right] \end{对齐}\begin{aligned} g\left(\Delta^{2}\right)=&-\frac{\bar{x}^{2}}{Q}\left(\frac{1}{n}+\ frac{\bar{x}^{2}}{Q}\right)^{-1}\left[G_{3, m}\left(\frac{1}{3} F_{1, m}( \alpha) ; \Delta^{2}\right)\right.\ &\left.-\Delta^{2}\left{2 G_{3, m}\left(\frac{1}{3} F_ {1, m}(\alpha) ; \Delta^{2}\right)-G_{5, m}\left(\frac{1}{5} F_{1, m}(\alpha) ; \Delta ^{2}\right)\right}\right] \end{对齐}
REff 的图表(一种,Δ2), 作为一个函数Δ2对于固定一种, 越过 1 线减少到最小值Δ2=Δ02(一种)（说）; 然后它向 1 线增加Δ2→∞. 的最大值REff⁡(一种,Δ2)发生在Δ2=0与价值
\operatorname{REff}(\alpha ; 0)=\left{1-\frac{\bar{x}^{2}}{Q}\left(\frac{1}{n}+\frac{\bar {x}^{2}}{Q}\right)^{-1} G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; 0\right )\right}^{-1} \geq 1,\operatorname{REff}(\alpha ; 0)=\left{1-\frac{\bar{x}^{2}}{Q}\left(\frac{1}{n}+\frac{\bar {x}^{2}}{Q}\right)^{-1} G_{3, m}\left(\frac{1}{3} F_{1, m}(\alpha) ; 0\right )\right}^{-1} \geq 1,

X=(19.383,21.117,18.99,19.415,20.394,20.212,20.163,20.521,20.125, 19.944,18.345,21.45,19.479,20.199,20.677,19.661,20.114,19.724 18.225,20.669)⊤

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