Probability plot for Cauchy Distribution

[herjia]

I have generated 2 columns of normal random variables, Z1 and Z2. Theorectically, Z1/Z2 will follow a Cauchy distribution. The question is, how do I construct a probability plot to show that indeed it is a Cauchy distribution?

I tried the follow procedure:
-Sort the Z1/Z2
-Rank them and store the rank on a new column, i
-perform median rank (herd-Johnson) i/n+1 where n is the sample size
-perform inverse cumulative probability on the median rank column
-plot the z1/z2 vs inverse cumulative probability

What i get is near the location, the data are linear while the deviation is serious at either extreme ends. any suggestion? or references?

 

[ZioX]

It's very very very important that the two standard normal variables you generated are independent. I don't know how you generated them so I don't know if they are.

It also goes without saying that you're going to need a sizeable amount of data to get anything meaningful.

Personally, I'd just prove that Z1/Z2 is a Cauchy distribution. It's fun!

 

[tacman]

A probability plot for a Cauchy distribution is a useful tool for visually assessing the fit of the data to the theoretical distribution. The procedure you have described is a commonly used method for constructing a probability plot. However, there are a few things to consider when interpreting the results.

First, it is important to note that the Cauchy distribution has heavy tails, meaning that extreme values are more likely to occur compared to a normal distribution. This can explain why you are seeing a linear relationship near the center of the plot, but deviations at the extreme ends. Therefore, it may be helpful to zoom in on the central portion of the plot to better assess the fit.

Another factor to consider is the sample size. The Cauchy distribution is known to have high variability, especially with smaller sample sizes. This can also contribute to the deviations you are seeing at the extreme ends of the plot. It may be helpful to increase the sample size to see if the plot becomes more linear.

Finally, it is always a good idea to compare the probability plot to a plot of the theoretical distribution. This can help you to assess the overall fit and identify any discrepancies.

In terms of references, there are many resources available online that discuss probability plots and their interpretation. You may also want to consult a statistics textbook or consult with a statistician for further guidance. Overall, it is important to keep in mind the characteristics of the Cauchy distribution when interpreting the results of your probability plot.