# Why is the Shapiro-Wilk $W$ statistic defined the way it is?

In the original paper (section 2.2.), we can see that derivation of the $$W$$ statistic goes along these lines:

1. Let let $$m_i$$ be the expected value of the $$i$$-th order statistic of a random sample of $$n$$ numbers from the standard normal distribution $$N(0, 1)$$ ($$i=1,2,\ldots,n$$) and let $$V$$ be the corresponding covariance matrix. Let $$m^T=[m_1,m_2,\ldots,m_n]$$.
2. If the ordered sample $$y^T=[y_1,y_2,\ldots,y_n]$$ comes from $$N(\mu,\sigma^2)$$ and the ordered sample $$[x_1,x_2,\ldots,x_n]$$ from $$N(0, 1)$$, then $$y_i=m_i+\sigma x_i$$ and the best linear unbiased estimator for $$\sigma$$ is $$\hat{\sigma}=\frac{m^TV^{-1}y}{m^TV^{-1}m}$$
3. In that case, it should hold that $$\hat{\sigma}^2=\frac{1}{n-1}\sum_{i=1}^{n}(y_i-\overline{y})^2$$, the usual estimator of variance.

Why wasn't the $$W$$ statistic defined simply as the corresponding ratio: $$\frac{\hat{\sigma}^2}{\frac{1}{n-1}\sum_{i=1}^{n}(y_i-\overline{y})^2}$$?

Instead, $$S^2=\sum_{1}^{n}(y_i-\overline{y})^2$$ is introduced as the estimator of $$(n-1)\sigma^2$$, and $$\hat{\sigma}^2$$ is multiplied by certain $$R^2=m^TV^{-1}m$$ and $$C^{-1}=(m^TV^{-1}V^{-1}m)^{-1/2}$$ to get $$b=\frac{R^2}{C}\hat{\sigma}^2=a^Ty\quad a^T=\frac{m^TV^{-1}}{(m^TV^{-1}V^{-1}m)^{1/2}}$$ so that $$W$$ could be defined as follows: $$W=\frac{b^2}{S^2}$$

So, my questions are:

1. Why bother with all those manipulations to get $$W$$ in that particular form?
2. What those multiplications and divisions mean? The authors say that $$C^2$$ is there to make sure that the linear coefficients are normalized, but what about multiplication with $$R^2$$ and why was the estimator of $$(n-1)\sigma^2$$ used? What it looks like to me is that $$R^2=m^TV^{-1}m$$ somehow corresponds to $$n-1$$ and that $$n-1$$ was deliberately removed so that the whole statistic doesn't depend on $$n$$.