Non-parametric single set test of the mean

[FunkyDwarf]

Hey gang,

I was wondering if there is a non-parametric version of the single set TTest? I know that often people refer to the Wilcoxon signed-rank test, but my understanding is that only tells you about the median, correct? Is there an equivalent that deals strictly with the mean?

Cheers!

 

[Greg Bernhardt]

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

 

[Stephen Tashi]

"Single sample T-test" instead of "Single set T-test"?

You should explain whether you are dealing with a particular problem or whether you are just stringing some words together to ask a theoretical question. Since the mean of a distribution is a parameter, how can we interpret the adjective "non-parametric" in a test "for the mean". What hyypothesis would be tested?

 

[FunkyDwarf]

I am dealing with a particular problem. I have a single set which I cannot assume is normally distributed, even when playing the transformation game. I would like to test whether or not the mean of this set is greater than zero in a way that is not dependent on some underlying distribution which has parameters. I would assume that given such a test exists for the median, admittedly not usually a parameter in distributions anyway, one would exist for the sample mean in the same manner, is this not correct?

 

[Stephen Tashi]

I know of no nonparametric hypothesis tests for the hypothesis that the mean of an arbitrary population is zero (or some other constant value). I think you need to assume some restrictions on the population distribution in order to have a nonparametric test. Most obviously, you need to assume the distribution has a mean value. If you assume the distribution is symmetric about the mean then there is something called a "sign test" that can be used.

( An interesting theoretical challenge would be to prove there is no nonparametric test of the hypothesis that the mean of a population is zero given only the assumption that the mean of the population distribution exists. It would be hard even to state what mathematical facts are to be proved! The question itself might need to be refined to rule out "meaningless" hypothesis tests. )

 

[FactChecker]

Whether your underlying distribution is known or not, the Central Limit Theorem says that the sample average will approach a normal distribution (given some assumptions that are very easy to meet). If you can get enough sample data, you can get sample estimates of the mean and variation and use the normal distribution.

 

[Number Nine]

Whether your underlying distribution is known or not, the Central Limit Theorem says that the sample average will approach a normal distribution (given some assumptions that are very easy to meet). If you can get enough sample data, you can get sample estimates of the mean and variation and use the normal distribution.

This is true, but it can be very hard to get enough data in practice, even if the mean and variance of the generating distribution exist. If your data are generated by some extremely fat tailed distribution (say, a t-distribution with just over 2-df), then the sampling distribution can be very fat-tailed even for sample sizes in the thousands. Any SE's or CI's derived under a normal assumption will be quite a ways off. It's not good practice to rely on the CLT to somehow "fix" a very non-normal sample.

 

[FactChecker]

It's not good practice to rely on the CLT to somehow "fix" a very non-normal sample.

I don't know of any other way to estimate the mean of a completely unknown distribution. Is there a better alternative?

 

[WWGD]

"Single sample T-test" instead of "Single set T-test"?

You should explain whether you are dealing with a particular problem or whether you are just stringing some words together to ask a theoretical question. Since the mean of a distribution is a parameter, how can we interpret the adjective "non-parametric" in a test "for the mean". What hyypothesis would be tested?

I think non-parametric just means that the general underlying distribution of the population is not know, i.e., it is not known whether the population involved is normal, t, F, x^2, etc. Of course, the parameters are not known, if they were, there would be non need for sampling.