Timeline for Testing the Anderson–Darling and Central Limit Theorem
Current License: CC BY-SA 4.0
11 events
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| Aug 3, 2023 at 13:23 | comment | added | Glen_b | 3. "That is, since the sample means of ANY distribution follow a normal distribution" This assertion is simply false... and it's not what the CLT says. | |
| Aug 3, 2023 at 13:22 | comment | added | Glen_b | 1. Testing is pointless; unless the original population distribution was already normal, the null is always false at any finite sample size. Why test null hypotheses guaranteed to be false? The reason why p-values don't seem to "settle" down is goodness of fit test's ability to pick up deviation from non-normality improves with sample size. 2. "effectiveness of the z-test and confidence intervals" Testing normality doesn't measure "effectiveness";' what matters for a test is accuracy of significance level and maintaining power. What matters for a CI is maintaining coverage and average width | |
| Aug 3, 2023 at 11:55 | comment | added | Kώστας Κούδας | @Glen_b What concerns me is that the effectiveness of the z-test and confidence intervals relies on the universality of the Central Limit Theorem (CLT). That is, since the sample means of ANY distribution follow a normal distribution, I can make an estimation of the population mean, regardless of its distribution. | |
| Aug 3, 2023 at 11:51 | comment | added | Kώστας Κούδας | @Knarpie I performed these tests for samples up to 630. The same issue persists. Moreover, one would expect that the p-values would increase as the sample size grows, but again, this is not happening. | |
| Aug 3, 2023 at 11:28 | comment | added | Glen_b | Even averages of one million observations doesn't get close to symmetry! See here: i.sstatic.net/KTMu0.png | |
| Aug 3, 2023 at 11:22 | comment | added | Glen_b |
@Kώστας If you believe that "n=30" is sufficient, try lognormal with $\sigma=4$ -- e.g. try x=replicate(10000,mean(rlnorm(1000,-5,4))). That's means of sample size 1000. Try: hist(x) and hist(log(x)). Notice that the log of the means is still somewhat right skew, even though the things being averaged must be symmetric on the log-scale. .... this is a distribution for which the actual CLT holds (which theorem says nothing whatever about n=30, or any other finite sample size). Then try replacing the sample size of 1000 with a sample size of 100000 (you'll want to drop the number of sims).
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| Aug 3, 2023 at 9:24 | comment | added | Knarpie | 30 is just a rule of thumb for data that are already close to normality anyway. Speed of convergence (i.e. the sample size needed to see normality) depends on many factors, e.g. on skewness of the data. Since you draw data from an exponential distribution, larger sample sizes may be needed. | |
| Aug 3, 2023 at 8:59 | comment | added | Kώστας Κούδας |
The Central Limit Theorem (CLT) can, I believe, be applied even to samples with a size of 30. In any case, I (if I'm not mistaken) start with samples of size s_s[1]=30 and end up with samples of size s_s[n_e]=30+n_e-1, which means a size of 100. Are these still considered small? Moreover, I don't see any differentiation in the p-values as I increase the sample sizes.
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| Aug 3, 2023 at 8:52 | comment | added | Knarpie | Because in the lower ranges of s_s, the sample size is too low for CLT, yielding non-normally distributed means. | |
| Aug 3, 2023 at 8:51 | comment | added | Kώστας Κούδας | You're right! I didn't notice that. Unfortunately, the problem persists even if I change s_s[1] to s_s[j]. | |
| Aug 3, 2023 at 8:42 | history | answered | Knarpie | CC BY-SA 4.0 |
