正態分佈、獨立性和方差齊次性檢驗
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正態分佈檢驗
經驗分布函數
\[\begin{aligned} \hat F_n(t) &=\frac{\text{number of elements in the sample}<t}{n} \\ &= \frac{1}{n} \sum_{i=1}^n 1_{x_i<t} \end{aligned}\]Shapiro-Wilk
Reference:
\[W=\frac{(\sum a_i x_{(i)})^2}{\sum(x_i-\bar x)^2}\]其中，
- $x_{(i)}$是從小到大排序後第$i$項
- ${Z_1, \cdots, Z_n}$ be i.i.d. $N(0, 1)$ and also take their order statistics $Z_{(1)}<\cdots<Z_{(n)} $
- $m_i$是$E(Z_{(i)})$
- $V$ 是 covariance of $Z_{(i)},Z_{(j)}$,即$V_{ij} = E[(Z_{(i)} −m_i)(Z_{(j)} − m_j )]$
- $C := (m’V^{−1}V^{−1}m)^{1/2}$
- $a=\frac{m’V^{-1}}{C}$
通常$W$越大，越接近正態分布 $H_0: X \text{ is normal distribution}$ 當 $W<W_\alpha$ 或者 $p < 1-\alpha $ 時，否定$H_0$
Kolmogorov-Smirnov
\[D=\text{Sup}| F_n(x)-F_0(x)|\]Cramer-von Mises
\[W^2=n \int_{-\infty}^{+\infty}(F_n(x)-F_0(x))^2dF_0(x)\]Anderson-Darling
\[A^2=n \int_{-\infty}^{+\infty}(F_n(x)-F_0(x))^2[F_0(x)(1-F_0(x))]^{-1}dF_0(x)\]Q-Q圖
繪制散點$(q_i,x^*_{(i)})$的散布圖，其中
\[\begin{aligned} q_i &=\Phi^{-1}(p_i) \\ p_i &=\frac{i-0.5}{n} \end{aligned}\]為正態總體的$p_i$分位數。
這些點應散布在一條$y=x$附近上。
獨立性檢驗
Durbin -Watson Test
\[d=\frac{\sum_{t=2}^T (e_t-e_{t-1})^2}{\sum_{t=1}^T e^2_t}\] \[H_0 :e_t= ae_{t-1}+v_t, a=0\]Since $d$ is approximately equal to
\[2(1 − \hat {\rho }),\]where $\hat {\rho }$ is the sample autocorrelation of the residuals, $d = 2$ indicates no autocorrelation.
A rule of thumb is that test statistic values in the range of 1.5 to 2.5 are relatively normal.
方差齊次性檢驗
Bartlett檢驗
\[\begin{aligned} s_i^2 &=\frac{1}{m_i-1} \sum_{j=1}^{m_i} (y_{ij}- \bar y_i)^2=\frac{Q_i}{f_i} \\ f_e&=f_1+f_2+\cdots+f_r=\sum_{i=1}^r(m_i-1) \\ \text{MS}_e &=\frac{1}{f_e}\sum_{i=1}^r Q_i=\sum_{i=1}^r \frac{f_i}{f_e}s_i^2\\ \text{GMS}_e &=[(s_1^2)^{f_1} (s_2^2)^{f_2} \cdots (s_r^2)^{f_r}]^{\frac{1}{f_e}}\\ \text{MS}_e &\geq\text{GMS}_e \\ C&=1+\frac{1}{3(r-1)}[\sum_{i=1}^r \frac{1}{f_i}-\frac{1}{f_e}]\\ B &=\frac{f_e}{C}( \text{ln MS}_e - \text{ln GMS}_e) \end{aligned}\]拒絕域為:
\[W=\{B \geq \chi_{(1-\alpha)}^2(r-1)\}\]