# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Robustness of bootstrap for testing means without normality and equal variances

(2015) STAM01 20151
Department of Statistics
Abstract
When testing for means there are two classical assumptions to be considered, normality and equal
variances (equal covariance matrices). Historically more attention seems to be directed towards
violation of the assumption of equal variances, which is referred to as the Behrens-Fisher problem,
than the assumption of normality. Our attention is equally directed to both the issues.
We propose a bootstrap based technique for testing means without the assumption of normality
and for the two-sample case also with unequal variances (unequal covariance matrices). The main
purpose of the thesis is to investigate the robustness of the bootstrap method for testing means,
both in the univariate and the multivariate case. The reason for interest... (More)
When testing for means there are two classical assumptions to be considered, normality and equal
variances (equal covariance matrices). Historically more attention seems to be directed towards
violation of the assumption of equal variances, which is referred to as the Behrens-Fisher problem,
than the assumption of normality. Our attention is equally directed to both the issues.
We propose a bootstrap based technique for testing means without the assumption of normality
and for the two-sample case also with unequal variances (unequal covariance matrices). The main
purpose of the thesis is to investigate the robustness of the bootstrap method for testing means,
both in the univariate and the multivariate case. The reason for interest in the multivariate case is
the relatively early stage of the research in this area. However, the purpose of the thesis is not a
complete study of the problem and therefore we impose some limitations such as using same sample
size and same distribution for both populations and excluding the two-sample multivariate case.
Initially we conclude that for all practical purposes the classical procedure and the bootstrap
method are equivalent in the benchmark case of normality and equal variances. We proceed by
investigating the performance of the approach in the cases in which the classical methods fail. The
general conclusion is that the method performs well for larger sample sizes, both in terms of signicance level and power. We also note some surprising and discrepant results. For instance, in
the univariate one-sample symmetric non-normal case the bootstrap method under some conditions
outperforms the test based on the true distribution by producing a correct signicance level and
higher power. Moreover, in some cases the bootstrap test outperforms the classical test in terms of
signicance level. This occurs in the one-sample asymmetric non-normal case (both univariate and
bivariate) and the univariate two-sample asymmetric non-normal case of equal variances. Finally, in
the univariate non-normal case of unequal variances we note an interesting result that the signicance
level appears to be quite wrong only for medium-sized samples.
To sum up, even if our approach may not perform well in some cases, it is still a decent method
that works quite well for dierent sample sizes, dierent distributions and dierent dimension of the
problem. However, we suggest that a more thorough investigation of the problem is needed. (Less)
author
supervisor
organization
course
STAM01 20151
year
type
H1 - Master's Degree (One Year)
subject
language
English
id
7456049
2015-06-30 10:10:44
date last changed
2015-06-30 10:10:44
```@misc{7456049,
abstract     = {When testing for means there are two classical assumptions to be considered, normality and equal
variances (equal covariance matrices). Historically more attention seems to be directed towards
violation of the assumption of equal variances, which is referred to as the Behrens-Fisher problem,
than the assumption of normality. Our attention is equally directed to both the issues.
We propose a bootstrap based technique for testing means without the assumption of normality
and for the two-sample case also with unequal variances (unequal covariance matrices). The main
purpose of the thesis is to investigate the robustness of the bootstrap method for testing means,
both in the univariate and the multivariate case. The reason for interest in the multivariate case is
the relatively early stage of the research in this area. However, the purpose of the thesis is not a
complete study of the problem and therefore we impose some limitations such as using same sample
size and same distribution for both populations and excluding the two-sample multivariate case.
Initially we conclude that for all practical purposes the classical procedure and the bootstrap
method are equivalent in the benchmark case of normality and equal variances. We proceed by
investigating the performance of the approach in the cases in which the classical methods fail. The
general conclusion is that the method performs well for larger sample sizes, both in terms of signicance level and power. We also note some surprising and discrepant results. For instance, in
the univariate one-sample symmetric non-normal case the bootstrap method under some conditions
outperforms the test based on the true distribution by producing a correct signicance level and
higher power. Moreover, in some cases the bootstrap test outperforms the classical test in terms of
signicance level. This occurs in the one-sample asymmetric non-normal case (both univariate and
bivariate) and the univariate two-sample asymmetric non-normal case of equal variances. Finally, in
the univariate non-normal case of unequal variances we note an interesting result that the signicance
level appears to be quite wrong only for medium-sized samples.
To sum up, even if our approach may not perform well in some cases, it is still a decent method
that works quite well for dierent sample sizes, dierent distributions and dierent dimension of the
problem. However, we suggest that a more thorough investigation of the problem is needed.},
author       = {Pejicic, Sasa},
language     = {eng},
note         = {Student Paper},
title        = {Robustness of bootstrap for testing means without normality and equal variances},
year         = {2015},
}

```