Robustness of Student t in the One-Sample Problem

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Level confidence intervals and -values are shown for 100 simple random samples of size drawn from the specified population with mean and standard deviation . The and methods are compared for calculating the confidence intervals and -values. The method is based on the normal distribution using the estimated standard deviation in place of the unknown . The method is approximately valid for large enough for all distributions considered in this Demonstration. The method uses the Student -distribution with degrees of freedom; it is exact in the case of the normal distribution and is approximate for the other distributions considered in this Demonstration.

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Corresponding to each confidence level for , the two-sided-value in the test of the null hypothesis (that the true mean equals 0) is shown to the right. These -values are uniformly distributed between 0 and 1 in the normal case using the method, and approximately so in the other cases.

It is interesting to do about 10,000 or more simulations to see how quickly the estimates and converge. To do this, click the icon beside the random seed slider, set the animation speed slower, and click the play button. Allow it to run for ten seconds or so.

This Demonstration can be used to show that the -test is not conservative in small samples. It also illustrates the random coverage probability of confidence intervals and the random distribution -values. Finally, the robustness of methods can be investigated for various non-normal distributions. See the Details for further discussion.

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Contributed by: Ian McLeod (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Each of the distributions used has population mean 0 and variance 1, as indicated on the scale at the bottom of the left panel. The -value is for testing versus . Of the three non-normal distributions considered, the methods work almost as well for the uniform and Laplace distributions as the normal for sample sizes as small as 10, but for such small samples the exponential distribution generates intervals and -tests that are not conservative. As increases, there is an improvement.

Snapshot 1: setting and using the -test method with the normal distribution and , we find after some simulations that the empirical values for the estimates and are close to their theoretical values of 0.95 and 0.05

Snapshot 2: as in Snapshot 1, but using the -test instead of the -test; after a large number of simulations, we see the empirical coverage probability is about 92%, which means the confidence intervals are too narrow; similarly the type I error at 8% is well above its nominal 5% value; this demonstrates that the -test method is not conservative in small samples

Snapshot 3: as in Snapshot 1, but with the uniform distribution; no observable difference from the normal population case

Snapshot 4: as in Snapshot 1, but with the Laplace distribution; this distribution has thicker tails than the normal, but even here the empirical coverage is not significantly different from the nominal 95% level, illustrating the robustness of the -distribution method

Snapshot 5: as in Snapshot 1, but with the exponential distribution; this distribution is right-skewed and very non-normal and in this case the method yields an empirical coverage rate of about % instead of the nominal 95% value; the method is not even conservative in this case

Snapshot 6: as in Snapshot 5, but the sample size is increased to ; the methods have improved and only slightly overstate the statistical significance of the test

Snapshot 7: as in Snapshot 6, but the sample size is increased to ; now the approximation is accurate



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