Testing for location after transformation to normality

Graduation date: 1989In the problem of testing the median using a random sample from a\ud certain distribution, and if no other parametric family is suggested,\ud the t-test is known to be the optimal procedure when this distribution\ud is normal. If the sample appears to be non-normal, one has the choice\ud either to consider a non-parametric approach or to try to correct for\ud non-normality before applying the t-test.\ud In this thesis we investigate the effect of applying certain power\ud transformations as an action to correct for non-normality before\ud applying the t-test. Also we investigate the effect of applying a\ud power transformation then trimming a certain proportion from the data\ud on each tail as a double action to correct for non-normality. This\ud problem is first considered by Doksum and Wong (1983), who apply the\ud Box-Cox power transformations to positive, right-skewed data when\ud testing for the equality of distributions of two independent samples.\ud In the present work we provide results for the one-sample case\ud using two alternatives to the Box-Cox power family which are applicable\ud to all data sets. Whenever it can be assumed that the data is a random\ud sample from a symmetric distribution with heavy tails, it is shown that\ud the John-Draper family of modtlus power transformations, with the\ud transformation parameter being positive and smaller than 1 , is\ud appropriate to correct for non-normality and the t-test based on the\ud transformed data is asymptotically more efficient and has better power\ud properties than the t-test based on the data in its original scale.\ud -When the data is thought to have a skewed distribution and can assume\ud negative as well as positive values, a new family of transformations,\ud referred to as the two-domain family, is introduced. It is shown that\ud the t-test based on the data after applying this new transformation is\ud also asymptotically more efficient and has better power properties than\ud the t-test in the original scale. A simulation study shows that\ud trimming a certain proportion on each tail of the data transformed by\ud one of the above two transformations then applying the t-test to the\ud trimmed samples yields a considerable gain in power compared to the\ud t-test in the original scale