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Q. Suppose for an experiment, the test for normality of an underlying population is accepted on the basis of a data set. To cross check the result, the experimener repeated the test with another independent sample, and now the test gets rejected. Discuss how the experimenter will come to the final conclusion based on these two findings.

I was solving test papers and found this question.

I know that we can use Pearsonian Chi square test to test Normality of a population , but can't figure out how to approach this question.

Please help.

Thank you

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There are a couple of possible answers to the question depending upon the circumstances. Often populations are assumed to be normal without anyone checking. Your question assumes two different groups looked, one failed to reject normality while the other did reject normality.

The first thing you could do is a meta-analysis, although I don't know of any on distributional assumptions that have been done. It is a giant literature, but I am not aware of any of that topic, personally.

The second thing is to look at the type of test used and the sample sizes. Small samples have less power. Likewise, the Shapiro-Wilks test has the greatest power, but the Kolmogorov-Smirnov test is robust. Check the properties of the specific testing procedures.

The third option is to notice that the normal distribution is a special case of the Pearson family of distributions. Rather than test to determine if normality can be rejected, test to see which member of the family is the best fit. You would use the elements of the first study to condition the second.

This would require a bit of work. You would need to use the first study to create a prior distribution for the second study. You have sampling data on the mean and variance, you might have data on skew and kurtosis. If you do not, you can assign the priors for them by performing a Monte Carlo simulation.

Given the sample size, test statistic, variance and mean estimators, how much skew and/or kurtosis would it take to push the test statistic into the rejection region. You know the sample estimate was less than that amount. You could assign a strong prior probability over the likely region and a small prior probability in the areas where rejection would have happened. You would then apply your data conditioned on the prior distribution from the first study.

You would want to then test your conclusions on the sensitivity of the result to the choice of prior distribution.

Alternatively, you could perform a meta-analysis on the fit to Pearson distributions, but I would look for an existing study that did just that. For the Bayesian variant, you could look at

Markowitz, H. and Usmen, N. (1996a). The likelihood of various stock market return distributions, part 1: principles of inference. Journal of Risk and Uncertainty, 13:207–219

and

Markowitz, H. and Usmen, N. (1996b). The likelihood of various stock market return distributions, part 2: empirical results. Journal of Risk and Uncertainty, 13:221–247.

You should not do what they did and first take the logarithmic transformation of the data as it changes the underlying distribution to map to its log counterpart. They did that to avoid negative values, but it creates a different type of issue that is as difficult to deal with.

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