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If we have a null hypothesis and subject it to Turning Point test, rank, runs portmant. etc. If the p value fall below 0.05 in some cases and not for others, do we then say that we reject the hypothesis?

I.e. what can we say about the WN hypothesis if our series passes some tests but fails at others?

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If you only do a few tests at a significance level that's not so high that you have a high chance of one or more type I errors ... and at least one is significant, then you would conclude that you don't have white noise on the basis of that rejection.

On the other hand, even if every test was not significant, it doesn't tell you you actually have white noise. You only have a failure to reject.

(Indeed, with real data, you don't need hypothesis tests to be confident it's not actually white noise. But assuming instead that the null could be actually true, every test done at say the 5% level carries a 5% chance of committing a type I error. So if you do enough tests, you can be certain some of them will give significant results just by chance. It's important to keep both these caveats in mind. In particular if you're using hypothesis tests to see if your data is suitable for some other procedure, they're not really answering the right question.)

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