Shapiro-Wilks test and the order statistic

In summary, the conversation discusses the interpretation of the order statistic in the context of the Shapiro-Wilk test. The participants confirm that if the data is sorted in ascending order, then the ith order statistic is equal to the ith element in the sequence. However, the Wiki page mentions that this may not always be the case. The conversation also mentions the convenience of finding the special case where the order statistic is equal to the element when computing the test.
  • #1
Mayhem
352
251
TL;DR Summary
How to interpret the order statistic in the context of SW
Given the Shapiro-Wilk test value W:
1654873563116.png

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?
 
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  • #2
Mayhem said:
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
 
  • #3
  • #5
Orodruin said:
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##.
 
  • #6
Fair enough.
 
  • #7
Orodruin said:
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.
 

FAQ: Shapiro-Wilks test and the order statistic

What is the Shapiro-Wilks test?

The Shapiro-Wilks test is a statistical test used to determine if a given data set follows a normal distribution. It is based on the comparison of the observed data with the expected values from a normal distribution. This test is commonly used in data analysis to check the assumption of normality before performing other statistical tests.

How does the Shapiro-Wilks test work?

The Shapiro-Wilks test works by calculating a test statistic, W, which measures the discrepancy between the observed data and the expected values from a normal distribution. This test statistic is then compared to a critical value from a table or calculated using statistical software. If the test statistic is less than the critical value, the data is considered to be normally distributed.

What is the purpose of using the Shapiro-Wilks test?

The Shapiro-Wilks test is used to determine if a data set follows a normal distribution. This is important because many statistical tests, such as t-tests and ANOVA, assume that the data is normally distributed. If the data is not normally distributed, these tests may produce inaccurate results. Therefore, the Shapiro-Wilks test is used to ensure the validity of these statistical tests.

What are the assumptions of the Shapiro-Wilks test?

The Shapiro-Wilks test assumes that the data is independent, continuous, and comes from a normally distributed population. Additionally, the test is most accurate when used with a sample size of at least 3 but less than 5000. If these assumptions are not met, the results of the Shapiro-Wilks test may not be reliable.

What is the relationship between the Shapiro-Wilks test and the order statistic?

The Shapiro-Wilks test uses the order statistic, which is the rank of each data point when the data is sorted in ascending order, to calculate the test statistic, W. The order statistic is used to determine the expected values from a normal distribution, which are then compared to the observed data to determine if the data is normally distributed. Therefore, the Shapiro-Wilks test and the order statistic are closely related in their calculation and interpretation.

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