# A test of whether a sample has a specified distribution

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:

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.

• 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. – semibruin Jul 22 '13 at 22:49
• @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? – Tim Jul 22 '13 at 22:53
• 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. – semibruin Jul 22 '13 at 22:56
• The aim here isn't to test normality (that's an assumption, not what's being tested). What is being tested here is autocorrelation. – Glen_b Jul 23 '13 at 0:30

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!