Testing the Mean of Skewed Distributions

Abstract

Johnson proposed some modified t tests to assess the mean of asymmetrical distributions. The tests were suggested for any parent distribution ranging from the normal to a distribution as asymmetric as an exponential distribution for sample sizes as small as 13. In many practical situations the skewness of the parent distribution can be greater than those studied by Johnson and the sample sizes can be quite small, possibly as small as 10, due to the cost of the sampling procedures. In these cases Johnson's test can be quite inaccurate. Sutton suggested a composite test to improve Johnson's upper-tailed t test. In this article a new test procedure is proposed for the upper-tailed test for the mean of positively skewed distributions, which requires only a little more effort than Johnson's test. A Monte Carlo study investigates the new procedure's properties for a variety of positively skewed distributions with small sample sizes. It is shown that the new test procedure is more accurate and more powerful than both Johnson's modified t test and Sutton's composite test.

Key Words:

• Composite test
• Edgeworth expansion
• Modified t test
• Power comparison
• Type I error rate