## 统计代写|回归分析作业代写Regression Analysis代考|STAT2220

2022年12月29日

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## 统计代写|回归分析作业代写Regression Analysis代考|EBRM and Uniqueness Conditions for MLEs

The $E B R M_W^3$ is presented in Definition $2.3$ and our study of this model in this section and the following sections includes fewer details than were provided for the $E B R M_B^3$ in the previous section. For definitions of the matrices used in Sects. 4.7$4.9$ the reader is referred to Chap. 3. The difference between the treatments of the two models will be highlighted, but usually only theorems containing the results are stated without proofs. Indeed, if one has followed the treatment of the $E B R M_B^3$, the proofs can be considered as classroom exercises. Estimators for the parameters of the $E B R M_W^3$ were given in Theorem 3.3, and in Corollary $3.4$ the estimator $\widehat{E[X]}$ was presented. As before, $\widehat{E[\boldsymbol{X}]}$ and $\widehat{\boldsymbol{\Sigma}}$ are always unique. When treating the $E B R M_B^3$, it was noted that the uniqueness of estimators is independent of the estimated inner product. Thus, by assuming $\boldsymbol{\Sigma}=\boldsymbol{I}$, the next theorem can be verified by transposing the matrices in the $E B R M_B^3$ and applying Theorem $4.9$.
Theorem 4.19 Consider the EBRM $M_W^3$ presented in Definition 2.3. Let $\widehat{\boldsymbol{B}}_i, i=$ $1,2,3$, be given in Theorem $3.3$ and let $\boldsymbol{K} \widehat{\boldsymbol{B}}_i \boldsymbol{L}, i=1,2,3$, be linear combinations of $\widehat{\boldsymbol{B}}_i ; \boldsymbol{K}$ and $\boldsymbol{L}$ are known matrices of proper sizes. Then the following statements hold:
(i) $\widehat{\boldsymbol{B}}_3$ is unique if and only if
$$r\left(\boldsymbol{A}_3\right)=q_3, \quad r\left(\boldsymbol{C}_3\right)=k_3, \quad \mathcal{C}\left(\boldsymbol{C}_3^{\prime}\right) \cap \mathcal{C}\left(\boldsymbol{C}_1^{\prime}: \boldsymbol{C}_2^{\prime}\right)={\mathbf{0}},$$
(ii) $\boldsymbol{K} \widehat{\boldsymbol{B}}_3 \boldsymbol{L}$ is unique if and only if
$$\mathcal{C}\left(\boldsymbol{K}^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{A}_3^{\prime}\right), \quad \mathcal{C}\left(\boldsymbol{L}^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}_3\left(\boldsymbol{C}_1^{\prime}: \boldsymbol{C}_2^{\prime}\right)^o\right) ;$$
(iii) $\widehat{\boldsymbol{B}}_2$ is unique if and only if
\begin{aligned} & r\left(\boldsymbol{A}_2\right)=q_2, \quad r\left(\boldsymbol{C}_2\right)=k_2, \quad \mathcal{C}\left(\boldsymbol{C}_1^{\prime}\right) \cap \mathcal{C}\left(\boldsymbol{C}_2^{\prime}\right)={\mathbf{0}} \ & \mathcal{C}\left(\boldsymbol{C}_1^{\prime}\right)^{\perp} \cap \mathcal{C}\left(\boldsymbol{C}_1^{\prime}: \boldsymbol{C}_2^{\prime}\right) \cap \mathcal{C}\left(\boldsymbol{C}_1^{\prime}: \boldsymbol{C}_3^{\prime}\right)={0} \end{aligned}
(iv) $\boldsymbol{K} \widehat{\boldsymbol{B}}_2 \boldsymbol{L}$ is unique if and only if
$$\mathcal{C}\left(\boldsymbol{K}^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{A}_2^{\prime}\right), \quad \mathcal{C}(\boldsymbol{L}) \subseteq \mathcal{C}\left(\boldsymbol{C}_2\left(\boldsymbol{C}_1^{\prime}: \boldsymbol{C}_3^{\prime}\right)^o\right)$$

## 统计代写|回归分析作业代写Regression Analysis代考|Asymptotic Properties of Estimators of Parameters

Concerning asymptotic properties we can once again rely completely on the approach and results for the $\angle B R M_B^3$. Correspondingly to Lemma $4.2$ the next lemma can be stated, whose proof follows directly from the proof of Lemma 4.2.
Lemma 4.5 Let $\boldsymbol{S}1, \widehat{\boldsymbol{S}}_2$ and $\widehat{\boldsymbol{S}}_3$ be as in Theorem 3.3. Suppose that for large n, $r\left(\boldsymbol{C}_1\right) \leq k_1$, and that both $r\left(\boldsymbol{C}_1: \boldsymbol{C}_2: \boldsymbol{C}_3\right)-r\left(\boldsymbol{C}_1: \boldsymbol{C}_2\right)$ and $r\left(\boldsymbol{C}_1: \boldsymbol{C}_2\right)-r\left(\boldsymbol{C}_1\right)$ do not depend on $n$. Then, if $n \rightarrow \infty$, $$n^{-1} \boldsymbol{S}_1 \stackrel{P}{\rightarrow} \mathbf{\Sigma}, \quad n^{-1} \widehat{\boldsymbol{S}}_2 \stackrel{P}{\rightarrow} \mathbf{\Sigma}, \quad n^{-1} \widehat{\boldsymbol{S}}_3 \stackrel{P}{\rightarrow} \mathbf{\Sigma} .$$ The following limiting quantities will be used: \begin{aligned} & \boldsymbol{K} \boldsymbol{B}{3 \Sigma} \boldsymbol{L}=\boldsymbol{K}\left(\boldsymbol{A}3^{\prime} \boldsymbol{\Sigma} \boldsymbol{A}_3\right)^{-} \boldsymbol{A}_3^{\prime} \boldsymbol{\Sigma} \boldsymbol{X} \boldsymbol{Q}_2 \boldsymbol{C}_3^{\prime}\left(\boldsymbol{C}_3 \boldsymbol{Q}_2 \boldsymbol{C}_3^{\prime}\right)^{-} \boldsymbol{L} \ & \boldsymbol{K} \boldsymbol{B}{2 \Sigma} \boldsymbol{L}=\boldsymbol{K}\left(\boldsymbol{A}2^{\prime} \boldsymbol{\Sigma}^{-1} \boldsymbol{A}_2\right)^{-} \boldsymbol{A}_2^{\prime} \boldsymbol{\Sigma}^{-1}\left(\boldsymbol{X}-\boldsymbol{A}_3 \boldsymbol{B}{3 \Sigma} \boldsymbol{C}3\right) \boldsymbol{Q}_1 \boldsymbol{C}_2^{\prime}\left(\boldsymbol{C}_2 \boldsymbol{Q}_1 \boldsymbol{C}_2^{\prime}\right)^{-} \boldsymbol{L},(4.156) \ & \boldsymbol{K} \boldsymbol{B}{1 \Sigma} \boldsymbol{L}=\boldsymbol{K}\left(\boldsymbol{A}1^{\prime} \boldsymbol{\Sigma}^{-1} \boldsymbol{A}_1\right)^{-} \boldsymbol{A}_1^{\prime} \boldsymbol{\Sigma}^{-1}\left(\boldsymbol{X}-\boldsymbol{A}_2 \boldsymbol{B}{2 \Sigma} \boldsymbol{C}2-\boldsymbol{A}_3 \boldsymbol{B}{3 \Sigma} \boldsymbol{C}3\right) \boldsymbol{C}_1^{\prime}\left(\boldsymbol{C}_1 \boldsymbol{C}_1^{\prime}\right)^{-} \boldsymbol{L} \end{aligned} which all are normally distributed, and where it is supposed that $\boldsymbol{K}$ and $\boldsymbol{L}$ are so chosen that (4.155)-(4.157) do not depend on the choice of $\mathrm{g}$-inverses, i.e. are unique. The matrices $Q_1$ and $Q_2$ are defined in (3.27), i.e. $Q_1=\boldsymbol{I}-\boldsymbol{P}{C_1^{\prime}}$ and $\boldsymbol{Q}2=\boldsymbol{I}-\boldsymbol{P}{C_1^{\prime}: C_2^{\prime}}$

Correspondingly to Theorem 4.10, where the $F R R M_B^3$ was considered, the next theorem can be verified.

# 回归分析代写

## 统计代写|回归分析作业代写Regression Analysis代考|EBRM and Uniqueness Conditions for MLEs

(i) $\widehat{\boldsymbol{B}}_3$ 是唯一的当且仅当
$$r\left(\boldsymbol{A}_3\right)=q_3, \quad r\left(\boldsymbol{C}_3\right)=k_3, \quad \mathcal{C}\left(\boldsymbol{C}_3^{\prime}\right) \cap \mathcal{C}\left(\boldsymbol{C}_1^{\prime}: \boldsymbol{C}_2^{\prime}\right)$$
(二) $K \widehat{B}_3 L$ 是唯一的当且仅当
$$\mathcal{C}\left(\boldsymbol{K}^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{A}_3^{\prime}\right), \quad \mathcal{C}\left(\boldsymbol{L}^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}_3\left(\boldsymbol{C}_1^{\prime}: \boldsymbol{C}_2^{\prime}\right)^{\circ}\right) ;$$
(三) $\widehat{\boldsymbol{B}}_2$ 是唯一的当且仅当
$$r\left(\boldsymbol{A}_2\right)=q_2, \quad r\left(\boldsymbol{C}_2\right)=k_2, \quad \mathcal{C}\left(\boldsymbol{C}_1^{\prime}\right) \cap \mathcal{C}\left(\boldsymbol{C}_2^{\prime}\right)=\mathbf{0}$$
(四) $K \widehat{B}_2 L$ 是唯一的当且仅当
$$\mathcal{C}\left(\boldsymbol{K}^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{A}_2^{\prime}\right), \quad \mathcal{C}(\boldsymbol{L}) \subseteq \mathcal{C}\left(\boldsymbol{C}_2\left(\boldsymbol{C}_1^{\prime}: \boldsymbol{C}_3^{\prime}\right)^o\right)$$

## 统计代写|回归分析作业代写Regression Analysis代考|Asymptotic Properties of Estimators of Parameters

$r\left(\boldsymbol{C}_1: \boldsymbol{C}_2: \boldsymbol{C}_3\right)-r\left(\boldsymbol{C}_1: \boldsymbol{C}_2\right)$ 和
$r\left(\boldsymbol{C}_1: \boldsymbol{C}_2\right)-r\left(\boldsymbol{C}_1\right)$ 不依赖 $n$. 那么，如果 $n \rightarrow \infty$ ，
$n^{-1} \boldsymbol{S}_1 \stackrel{P}{\rightarrow} \boldsymbol{\Sigma}, \quad n^{-1} \widehat{\boldsymbol{S}}_2 \stackrel{P}{\rightarrow} \boldsymbol{\Sigma}, \quad n^{-1} \widehat{\boldsymbol{S}}_3 \stackrel{P}{\rightarrow} \boldsymbol{\Sigma}$.

$\boldsymbol{K} \boldsymbol{B} 3 \Sigma \boldsymbol{L}=\boldsymbol{K}\left(\boldsymbol{A} 3^{\prime} \boldsymbol{\Sigma} \boldsymbol{A}_3\right)^{-} \boldsymbol{A}_3^{\prime} \boldsymbol{\Sigma} \boldsymbol{X} \boldsymbol{Q}_2 \boldsymbol{C}_3^{\prime}\left(\boldsymbol{C}_3 \boldsymbol{Q}_2 \boldsymbol{C}_3^{\prime}\right)$

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