## 统计代写|回归分析作业代写Regression Analysis代考|STA321

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|Asymptotic Properties of Estimators of Parameters

Similar to Lemma $4.1$ the next lemma can be established.
Lemma 4.2 Let $\boldsymbol{S}1, \widehat{\boldsymbol{S}}_2, \widehat{\boldsymbol{S}}_3, \widehat{\boldsymbol{Q}}_1, \widehat{\boldsymbol{Q}}_2, \boldsymbol{Q}_1$ and $\boldsymbol{Q}_2$ be defined through Theorem $3.2$ and (3.13)-(3.16). Suppose that for large n, $r\left(\boldsymbol{C}_1\right) \leq k_1$, and that both $r\left(\boldsymbol{C}_1\right)-$ $r\left(\boldsymbol{C}_2\right)$ and $r\left(\boldsymbol{C}_2\right)-r\left(\boldsymbol{C}_3\right)$ are independent of $n$. Then, as $n \rightarrow \infty$, (i) $n^{-1} \boldsymbol{S}_1 \stackrel{P}{\rightarrow} \mathbf{\Sigma}, \quad n^{-1} \widehat{\boldsymbol{S}}_2 \stackrel{P}{\rightarrow} \mathbf{\Sigma}, \quad n^{-1} \widehat{\boldsymbol{S}}_3 \stackrel{P}{\rightarrow} \mathbf{\Sigma}$ (ii) $\widehat{\boldsymbol{Q}}_1 \stackrel{P}{\rightarrow} \boldsymbol{Q}_1, \quad \widehat{\boldsymbol{Q}}_2 \stackrel{P}{\rightarrow} \boldsymbol{Q}_2$ Proof Since the distribution for $\boldsymbol{S}$ (see Lemma 4.1) used in the BRM and the distribution for $S_1$ are the same, $n^{-1} S_1 \stackrel{P}{\rightarrow} \Sigma$ follows from Lemma 4.1, and this is also true for $\widehat{\boldsymbol{Q}}_1 \stackrel{P}{\rightarrow} \boldsymbol{Q}_1$. Then it is noted that $\widehat{\boldsymbol{Q}}_1^{\prime} \boldsymbol{A}_1=\mathbf{0}$, and hence $$\widehat{\boldsymbol{S}}_2=\boldsymbol{S}_1+\widehat{\boldsymbol{Q}}_1^{\prime}\left(\boldsymbol{X}-\boldsymbol{A}_1 \boldsymbol{B}_1 \boldsymbol{C}_1\right)\left(\boldsymbol{P}{C_1^{\prime}}-\boldsymbol{P}{C_2^{\prime}}\right)\left(\boldsymbol{X}-\boldsymbol{A}_1 \boldsymbol{B}_1 \boldsymbol{C}_1\right)^{\prime} \widehat{\boldsymbol{Q}}_1 .$$ From Appendix B, Theorem B.20 (vi) it follows that $$\left(\boldsymbol{X}-\boldsymbol{A}_1 \boldsymbol{B}_1 \boldsymbol{C}_1\right)\left(\boldsymbol{P}{C_1^{\prime}}-\boldsymbol{P}{C_2^{\prime}}\right)\left(\boldsymbol{X}-\boldsymbol{A}_1 \boldsymbol{B}_1 \boldsymbol{C}_1\right)^{\prime} \sim W_p\left(\boldsymbol{\Sigma}, r\left(\boldsymbol{C}_1\right)-r\left(\boldsymbol{C}_2\right)\right),$$ because $\left(\boldsymbol{A}_3 \boldsymbol{B}_3 \boldsymbol{C}_3+\boldsymbol{A}_2 \boldsymbol{B}_2 \boldsymbol{C}_2\right)\left(\boldsymbol{P}{C_1^{\prime}}-\boldsymbol{P}{C_2^{\prime}}\right)=\mathbf{0}$. It is assumed that $r\left(\boldsymbol{C}_1\right)-r\left(\boldsymbol{C}_2\right)$ is fixed for large $n$, which indeed implies that for large $n$ the Wishart distribution does not depend on the values of $n$. Hence, $$\frac{1}{n}\left(\boldsymbol{X}-\boldsymbol{A}_1 \boldsymbol{B}_1 \boldsymbol{C}_1\right)\left(\boldsymbol{P}{C_1^{\prime}}-\boldsymbol{P}_{C_2^{\prime}}\right)\left(\boldsymbol{X}-\boldsymbol{A}_1 \boldsymbol{B}_1 \boldsymbol{C}_1\right)^{\prime} \stackrel{P}{\rightarrow} 0,$$
which is precisely what is needed in the following. Thus, (4.43) yields $n^{-1}$ ( $\widehat{\boldsymbol{S}}_2-$ $\left.\boldsymbol{S}_1\right) \stackrel{P}{\rightarrow} \mathbf{0}$, and then $n^{-1} \widehat{\boldsymbol{S}}_2 \stackrel{P}{\rightarrow} \mathbf{\Sigma}$. Moreover, $\widehat{\boldsymbol{Q}}_2 \stackrel{P}{\rightarrow} \boldsymbol{Q}_2$ and then copying the above presentation one may show $n^{-1} \widehat{\boldsymbol{S}}_3 \stackrel{P}{\rightarrow} \boldsymbol{\Sigma}$.

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

For the $B R M$, the distributions of the maximum likelihood estimators are difficult to find. In Theorem 3.2, the estimators for the $E B R M_B^3$ were given and one can see that the expressions are stochastically much more complicated than the estimators for the $B R M$. To understand the estimators, moments are useful quantities. For example, approximations of the distributions of the estimators have to take place, and in this book these approximations are based on moments. Before studying $\boldsymbol{K} \widehat{\boldsymbol{B}}i \boldsymbol{L}, i=1,2,3$, the estimated mean structure $\widehat{E[\boldsymbol{X}]}=\sum{i=1}^3 \boldsymbol{A}_i \widehat{\boldsymbol{B}}_i \boldsymbol{C}_i$ and $\widehat{\boldsymbol{\Sigma}}$ are treated. Thereafter, $D\left[\boldsymbol{K} \widehat{\boldsymbol{B}}_i \boldsymbol{L}\right], i=1,2,3$, is calculated. The ideas for calculating $D\left[\boldsymbol{K} \widehat{\boldsymbol{B}}_i \boldsymbol{L}\right]$ are very similar to the ones presented for obtaining $D[\widehat{\boldsymbol{E}[\boldsymbol{X}]}]$ and $E[\widehat{\boldsymbol{\Sigma}}]$. Some advice is appropriate here. The technical treatment in this section is complicated, although not very difficult. Readers less interested in details are recommended merely to study the results in the given theorems. Moreover, the presentation in different places is not complete due to computational lengthiness. Table $4.1$ includes definitions which are used throughout the section.

First it will be shown that in the $E B R M_B^3$, under the uniqueness conditions presented in Theorem 4.9, the maximum likelihood estimators of $\boldsymbol{K} \boldsymbol{B}i \boldsymbol{L}$ will be unbiased and then it follows that $\widehat{E[X]}=\sum{i=1}^m \boldsymbol{A}_i \widehat{\boldsymbol{B}}_i \boldsymbol{C}_i$ is also unbiased. In Theorem $3.2$ the maximum likelihood estimators $\widehat{\boldsymbol{B}}_i, i=1,2,3$, were presented. Since $\mathcal{C}\left(\boldsymbol{C}_3^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}_2^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}_1^{\prime}\right)$, the following facts, which are obtained from Appendix B, Theorem B.19 (ix) and (xi), will be utilized.

# 回归分析代写

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## MATLAB代写

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