I Shapiro-Wilks test and the order statistic

[Mayhem]

TL;DR Summary

How to interpret the order statistic in the context of SW

Given the Shapiro-Wilk test value W:

where I'm interested in the numerator. If my data is sorted in ascending order, my understanding is that $x_(i) = x_i$. Is that correct?

 

[Mark44]

Summary: How to interpret the order statistic in the context of SW

Given the Shapiro-Wilk test value W:
View attachment 302652
where I'm interested in the numerator. If my data is sorted in ascending order, my understanding is that $x_(i) = x_i$. Is that correct?

Sounds right to me. From the wiki page, https://en.wikipedia.org/wiki/Shapiro–Wilk_test:

$x_{(i)}$ (with parentheses enclosing the subscript index i; not to be confused with $x_i$) is the ith order statistic, i.e., the ith-smallest number in the sample

 

[Mayhem]

Sounds right to me. From the wiki page, https://en.wikipedia.org/wiki/Shapiro–Wilk_test:

Thanks. I just wanted to be sure, as I was implementing the formula in Python.

 

[Orodruin]

Sounds right to me. From the wiki page, https://en.wikipedia.org/wiki/Shapiro–Wilk_test:

The Wiki page seems to be saying explicitly that in general $x_{(i)} \neq x_i$.

 

[Mark44]

The Wiki page seems to be saying explicitly that in general $x_{(i)} \neq x_i$.

But if the elements of the sequence are already ordered from smallest to largest, then $x_{(i)} = x_i$.

 

[Orodruin]

Fair enough.

 

[Mayhem]

The Wiki page seems to be saying explicitly that in general $x_{(i)} \neq x_i$.

Yes, but for computation, finding the special case where $x_{(i)} = x_i$ makes life easier, which I did, and I got the right results when debugging random samples against known calculators.