统计代写|应用线性模型代写Applied Linear Models代考|STAT501

统计代写|应用线性模型代写Applied Linear Models代考|RELATED TOPICS

It is appropriate to briefly mention certain topics related to the preceding development that are customarily associated with testing hypotheses. The treatment of these topics will do no more than act as an outline to the reader, showing him their application to the linear models situation. As with the discussion of distribution functions in Chapter 2, the reader will have to look elsewhere for a complete discussion of these topics.
a. The likelihood ratio test
Tests of linear hypotheses $\mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}$ have been developed from the starting point of the $F$-statistic. This, in turn, can be shown to stem from the likelihood ratio test.

For a sample of $N$ observations $\mathbf{y}$, where $\mathbf{y}$ is $N\left(\mathbf{X b}, \sigma^2 \mathbf{I}\right)$ the likelihood function is
$$L\left(\mathbf{b}, \sigma^2\right)=\left(2 \pi \sigma^2\right)^{-\frac{1}{2} N} \exp \left{-\left[(\mathbf{y}-\mathbf{X b})^{\prime}(\mathbf{y}-\mathbf{X b}) / 2 \sigma^2\right]\right} .$$
The likelihood ratio test utilizes two values of $L\left(\mathbf{b}, \sigma^2\right)$ :
(i) $\operatorname{Max}\left(L_w\right)$, the maximum value of $L\left(\mathbf{b}, \sigma^2\right)$ maximized over the complete range of parameters, namely $0<\sigma^2<\infty$, and $-\infty<b_i<\infty$ for all $i$.
(ii) $\operatorname{Max}\left(L_H\right)$, the maximum value of $L\left(\mathbf{b}, \sigma^2\right)$ maximized over the range of parameters limited (restricted or defined) by the hypothesis $H$.
The likelihood ratio is the ratio of these two maxima:
$$L=\frac{\max \left(L_H\right)}{\max \left(L_w\right)} .$$
Each maximum is found in the usual manner: differentiate $L\left(\mathbf{b}, \sigma^2\right)$ with respect to $\sigma^2$ and the elements of $\mathbf{b}$, equate the differentials to zero, solve the resulting equations for $\mathbf{b}$ and $\sigma^2$ and use these solutions in the place of $\mathbf{b}$ and $\sigma^2$ in $L\left(\mathbf{b}, \sigma^2\right)$. In the case of $\max \left(L_H\right)$ the maximization procedure is carried out within the limitations of the hypothesis. We demonstrate for the case of the hypothesis $H: \mathbf{b}=\mathbf{0}$. First, $\partial L\left(\mathbf{b}, \sigma^2\right) / \partial \mathbf{b}=\mathbf{0}$ gives, as we have seen, $\hat{\mathbf{b}}=\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \mathbf{y} ;$ and $\partial L\left(\mathbf{b}, \sigma^2\right) / \partial \sigma^2=0$ gives $\hat{\sigma}^2=(\mathbf{y}-\mathbf{X} \hat{\mathbf{b}})^{\prime}(\mathbf{y}-\mathbf{X} \hat{\mathbf{b}}) / N$

统计代写|应用线性模型代写Applied Linear Models代考|Type I and II errors

Under the null hypothesis $H: \mathbf{K}^{\prime} \mathbf{b}=\mathbf{m}, F(H)=(N-r) Q / s$ SSE has the $F_{s, N-r}$ distribution. For a significance test at the $100 \alpha \%$ level the rule of the test is to not reject $H$ whenever $F(H) \leq F_{\alpha, s, N-r}$, the tabulated value of the $F_{s, N-r}$ distribution, at the $100 \alpha \%$ point. This means $F_{\alpha, s, N-r}$ is defined as follows: if $u$ is any variable having the $F_{s, N-r}$ distribution then
$$\operatorname{Pr}\left{u \geq F_{\alpha, s, N-r}\right}=\alpha .$$
The probability $\alpha$ is the (significance) level of the significance test. An oftused value for it is $0.05$, but there is nothing sacrosanct about this; any value between 0 and 1 can be used for $\alpha$. Other frequently used values are $0.01$ and $0.10$.

The rule of whether or not to reject the hypothesis $H$ is to reject it whenever $F(H)>F_{\alpha, s, N-r}$ and to not reject it whenever $F(H) \leq F_{\alpha, s, N-r}$. By the nature of the statistic $F(H)$ we know that over repeated sampling $F(H)$ will exceed $F_{\alpha, s, N-r}$ on $100 \alpha \%(5 \%$, say $)$ of the time; and when it does we will reject $H$. Therefore, in situations in which the null hypothesis $H$ is actually true, this rejection will constitute an error of judgment. It is the error known as a Type I error, or rejection error. It consists of wrongly rejecting the null hypothesis $H$ when it is true; the probability of its occurrence is $\alpha$.

统计代写|应用线性模型代写Applied Linear Models代考|RELATED TOPICS

(二) $\operatorname{Max}\left(L_H\right)$ ，最大值 $L\left(\mathbf{b}, \sigma^2\right)$ 在假设限制（限制或定义) 的参数范围内最大化 $H$. 似然比是这两个最大值的比值:
$$L=\frac{\max \left(L_H\right)}{\max \left(L_w\right)} .$$

统计代写|应用线性模型代写Applied Linear Models代考|Type I and II errors

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