Abstract
It is important to examine the symmetry of an underlying distribution before applying some statistical procedures to a data set. For example, in the Zuni School District case, a formula originally developed by the Department of Education trimmed 5% of the data symmetrically from each end. The validity of this procedure was questioned at the hearing by Chief Justice Roberts. Most tests of symmetry (even nonparametric ones) are not distribution free in finite sample sizes. Hence, using asymptotic distribution may not yield an accurate type I error rate or/and loss of power in small samples. Bootstrap resampling from a symmetric empirical distribution function fitted to the data is proposed to improve the accuracy of the calculated p-value of several tests of symmetry. The results show that the bootstrap method is superior to previously used approaches relying on the asymptotic distribution of the tests that assumed the data come from a normal distribution. Incorporating the bootstrap estimate in a recently proposed test due to Miao, Gel and Gastwirth (2006) preserved its level and shows it has reasonable power properties on the family of distribution evaluated.
Highlights
As noted by Lehmann and Romano (2005, page 248) the problem of testing whether data comes from a symmetric distribution when the center is unknown is more difficult than the corresponding problem when the center is known
In the process of calculating the measure of relative disparity used to determine whether the expenditures in the school districts of a state are sufficiently equal for a state, rather than the local districts, to receive most of the Federal Impact Aid money, the Department of Education deletes the largest and smallest 5% of the data
This paper shows that the sampling distribution of several tests of symmetry can be estimated using the bootstrap when re-samples are taken from the symmetrized empirical CDF about the sample median
Summary
As noted by Lehmann and Romano (2005, page 248) the problem of testing whether data comes from a symmetric distribution when the center is unknown is more difficult than the corresponding problem when the center is known. The Justice asked why none of the three parties involved in the case discussed the issue of outliers in their briefs His question is quite important as trimming the upper and lower 5 or 10% of the data is a well accepted method when the objective is to estimate the center of the data The parameters of the limiting distribution can be estimated, educational funding data typically refers to school districts in a state.
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