Log-Rank vs X Test for Exponentiality

This paper appraises a convenient test sometimes recommended to determine whether a set of observations has been drawn from an exponential distribution with unknown mean. The test uses simple linear regression techniques. Historically, it has been used in an intuitive manner. The intuitive procedure usually involves plotting logarithms of the empirical Cdf against corresponding observed values, then `eyeballing' the plotted points for linearity, or intuitively determining whether r2 calculated for the bivariate distribution is `high enough' or not. Using the objective procedure introduced in this paper, one regresses logarithms of ranks against observed values, calculates a standardized slope statistic, and checks this value against the tabled rejection region(s) provided. Our appraisal of the s-power of the objective log-rank test suggests that it is less s-powerful than competing tests (W, S*, D*) at larger sample sizes. Its relative performance appears to improve somewhat for smaller sample sizes. It seems fair to describe the objective log-rank test as a medium-grade test. Therefore, the practitioner should use the competing tests, unless samples are small, or practical considerations, such as convenience, are decisive in some particular situation. If convenience is important, then the log-rank test with the standardized slope used as the test statistic is an attractive option. The use of the log-rank test in its intuitive form is not recommended at all, since it very likely inclines the practitioner too often to accept the exponential hypothesis when false.