## 统计代写|应用线性模型代写Applied Linear Models代考|Estimation under the null hypothesis

When considering the hypothesis $H: \quad \mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}$ it is natural to ask, “What is the estimator of $\mathbf{b}$ under the null hypothesis?” This might be especially pertinent following non-rejection of the hypothesis by the preceding $F$-test. The desired estimator, $\tilde{b}$ say, is readily obtainable using constrained least squares. Thus $\tilde{\mathbf{b}}$ is derived so as to minimize $(\mathbf{y}-\mathbf{X} \tilde{\mathbf{b}})^{\prime}(\mathbf{y}-\mathbf{X} \tilde{\mathbf{b}})$ subject to the constraint $\mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}$.

With $2 \theta^{\prime}$ as a vector of Lagrange multipliers we minimize
$$(\mathbf{y}-\mathbf{X} \tilde{b})^{\prime}(\mathbf{y}-\mathbf{X} \tilde{\mathbf{b}})+2 \mathbf{\theta}^{\prime}\left(\mathbf{K}^{\prime} \tilde{\mathbf{b}}-\mathbf{m}\right)$$
with respect to the elements of $\overrightarrow{\mathbf{b}}$ and $\boldsymbol{\theta}$. Differentiation with respect to these elements leads to the equations
with the other terms vanishing because $\mathbf{X}^{\prime}(\mathbf{y}-\mathbf{X} \hat{\mathbf{b}})=\mathbf{0}$.

## 统计代写|应用线性模型代写Applied Linear Models代考|Reduced models

We now consider, in turn, the effect on the model $\mathbf{y}=\mathbf{X b}+\mathbf{e}$ of the hypotheses $\mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}, \mathbf{K}^{\prime} \mathbf{b}=\mathbf{0}$ and $\mathbf{b}q=\mathbf{0}$. (i) $\mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}$. In estimating $\mathbf{b}$ subject to $\mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}$ it could be said that we are dealing with a model $\mathbf{y}=\mathbf{X b}+\mathbf{e}$ on which has been imposed the limitation $\mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}$. We refer to the model that we start with, $\mathbf{y}=\mathbf{X b}+\mathbf{e}$ without the limitation, as the full model; and the model with the limitation imposed, $\mathbf{y}=\mathbf{X b}+\mathbf{e}$ with $\mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}$, is called the reduced model. For example, if the full model is $$y_i=b_0+b_1 x{i 1}+b_2 x_{i 2}+b_3 x_{i 3}+e_i$$
and the hypothesis is $H: b_1=b_2$, the reduced model is
$$y_i=b_0+b_1\left(x_{i 1}+x_{i 2}\right)+b_3 x_{i 3}+e_i .$$
The meaning of $Q$ and of SSE $+Q$ is now investigated in terms of sums of squares associated with the full and reduced models. To aid description we introduce the terms reduction(full) and residual(full) for the reduction and residual sums of squares after fitting the full model:
reduction(full) $=$ SSR $\quad$ and $\quad$ residual(full $)=$ SSE.
Similarly
$$\mathrm{SSE}+Q=\text { residual(reduced), }$$
as established in (105). Hence
\begin{aligned} Q &=\mathrm{SSE}+Q-\mathrm{SSE} \ &=\text { residual(reduced })-\text { residual(full) } \end{aligned} and also
\begin{aligned} Q &=\mathbf{y}^{\prime} \mathbf{y}-\mathrm{SSE}-\left[\mathbf{y}^{\prime} \mathbf{y}-(\mathrm{SSE}+Q)\right] \ &=\mathrm{SSR}-\left[\mathbf{y}^{\prime} \mathbf{y}-(\mathrm{SSE}+Q)\right] \ &=\text { reduction(full })-\left[\mathbf{y}^{\prime} \mathbf{y}-(\mathrm{SSE}+Q)\right] \end{aligned}

## 统计代写|应用线性模型代写Applied Linear Models代考|Estimation under the null hypothesis

$$(\mathbf{y}-\mathbf{X} \tilde{b})^{\prime}(\mathbf{y}-\mathbf{X} \tilde{\mathbf{b}})+2 \theta^{\prime}\left(\mathbf{K}^{\prime} \tilde{\mathbf{b}}-\mathbf{m}\right)$$

## 统计代写|应用线性模型代写Applied Linear Models代考|Reduced models

$$y_i=b_0+b_1 x i 1+b_2 x_{i 2}+b_3 x_{i 3}+e_i$$

$$y_i=b_0+b_1\left(x_{i 1}+x_{i 2}\right)+b_3 x_{i 3}+e_i .$$

$$\mathrm{SSE}+Q=\text { residual(reduced) }$$

$$Q=\mathrm{SSE}+Q-\mathrm{SSE} \quad=\text { residual(reduced })-\operatorname{residual}(\text { full })$$

$$\left.Q=\mathbf{y}^{\prime} \mathbf{y}-\operatorname{SSE}-\left[\mathbf{y}^{\prime} \mathbf{y}-(\mathrm{SSE}+Q)\right] \quad=\operatorname{SSR}-\left[\mathbf{y}^{\prime} \mathbf{y}-(\mathrm{SSE}+Q)\right]=\text { reduction(full }\right)-\left[\mathbf{y}^{\prime} \mathbf{y}\right.$$

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