Testing Mean Returns Under a Stable Distribution

Summary

The document considers whether a conventional t-test is suitable for testing the significance of trading returns when returns are modeled with a stable distribution. It points readers toward a paper on a z-test for the mean of a stable probability distribution, which is presented as a possible alternative to a test based on normality assumptions.

The responses also note that stable-distribution parameters can be estimated by maximum likelihood and that software packages can provide parameter estimates and confidence intervals. These suggestions offer a route for modeling the return distribution and quantifying uncertainty, but the document does not explain the z-test’s assumptions, derivation, or finite-sample behavior. It supplies no worked example or comparison of test performance. In particular, readers would need to verify whether the selected stable model and inference method fit their data and research question before relying on a significance result.

Key ideas

• The question is whether a conventional t-test fits returns modeled with a stable distribution.
• A z-test for the mean of a stable distribution is suggested as a potential alternative.
• Stable-distribution parameters can be estimated using maximum likelihood.
• Parameter confidence intervals can help describe estimation uncertainty.
• The discussion gives references but no derivation, worked example, or validation of the suggested test.

Tags

# Significance of return under stable distribution


# Significance of return under stable distribution












if I want to use `t-test` to test significance of my returns, it assumes the random variable is distributed normally. But in my work I work under stable distributed returns. It seems inappropriate to use this test then. Is there an alternative? Cannot google anything.

Cheers guys

## Answer by nbbo2 (score 0, accepted)

https://quant.stackexchange.com/a/30048

I am not an expert on this topic, but I saw a recent article

Parkinson (2102): z Test for the significance of the mean of a stable probability distribution http://tandfonline.com/doi/abs/10.1080/02664763.2012.740618

that may be of interest to you.

## Answer by user1483 (score 0)

https://quant.stackexchange.com/a/30143

The four parameters of a stable distribution may be estimated by maximum likelihood. John Nolan's stable package is available from http://www.robustanalysis.com/ in forms suitable for use in C, C++, R, matlab Mathematica and excel with Windows, Linux or Mac. The package estimates the parameters and produces confidence intervals.

You might also look at the stable.exe program available at http://fs2.american.edu/jpnolan/www/stable/stable.html. This also does maximum likelihood estimates of the parameters and produces confidence intervals.

You might also find some of the material in http://www.tcd.ie/Economics/staff/frainj/Stable_Distribution/thesis_main_5.pdf useful

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.