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Suppose we have an iid sample $X_1, \dots, X_n$. We want to test if it has a standard normal distribution.

One test statistic I learned from Brockwell and Davis's Introduction to Time Series and Forecasting is to

  • first find an interval $(-b, b)$ which has a probability $95\%$ under the standard normal distribution,
  • then compute the ratio $r$ of the sample points falling into $(-b,b)$.

The book continues to say if $r$ is not equal to $95\%$, then reject the null. But I think it is not right. What I think should be done instead is to find the distribution of $r$ under the null (and then find the rejection region of $r$). but I am not sure how to do that?

This test should have very low power,isn't it? Because the test statistic only captures a little information about the sample's distribution?

Thanks and regards!

PS: in case my understanding is incorrect, the original text from the book says:

enter image description here

Note that the book tells how to test if a sample is iid, but its underlying idea is to test if the sample autocorrelations is iid with distribution $N(0,1/n)$, which is the same question as in my post.

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    $\begingroup$ The specification test your mentioned is a special case of Pearson's chi-square test. There are many alternative goodness-of-fit tests. You may be also interested in Kolmogorov-Smirnov test. $\endgroup$ – semibruin Jul 22 '13 at 22:49
  • $\begingroup$ @semibruin: Thanks! This test should have very low power,isn't it? Because the interval only captures a little information about the sample's distribution? $\endgroup$ – Tim Jul 22 '13 at 22:53
  • $\begingroup$ Yes, your intuition is right. Its power is low. Also, the test is sorta arbitrary in the sense that the selection of the confidence interval could be arbitrary. $\endgroup$ – semibruin Jul 22 '13 at 22:56
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    $\begingroup$ The aim here isn't to test normality (that's an assumption, not what's being tested). What is being tested here is autocorrelation. $\endgroup$ – Glen_b Jul 23 '13 at 0:30
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The initially proposed test is a special case of the chisquared goodness of fit test. Much discussion on this site, see this search list. You could also look into the Kolmogorov-Smirnov test.

But, your question is not really about goodness of fit to a normal distribution, it is about testing the autocorrelation function. The test given, while useful, is approximate and more to be thought about as a visual "guide for the eye". Even under the null of no autocorrelation, the individual autocorrelations at different lags will not be independent, so the assumption of independence used can only be an approximation. And yes, better, more powerful tests are possible.

An example is in the R package forecast with functions Acf and taperedacf which optionally can plot simulation-based confidence intervals for the autocorrelations. But using much more computational power!

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