Distinguishing between the exponential and Lindley distributions: An illustration from biological psychology

The exponential distribution has been used for modeling positively skewed data in biological psychology. However, the lesser-known Lindley distribution, although not typically used for this purpose, has a density and cumulative distribution that are very similar to those of the exponential distribution. This similarity suggests that the Lindley distribution could be a strong candidate for modeling such data types. While the probability density and cumulative distribution functions of these two one-parameter distributions can be quite similar, their hazard rate functions differ markedly. Therefore, selecting the most appropriate distribution significantly impacts the interpretation of the hazard rate function. To aid in this selection, we introduce a method that distinguishes between the exponential and Lindley distributions by examining the ratio of their maximized likelihood functions. This method is versatile, as it can also be applied to type I censored data, enhancing its practical appeal. Asymptotic results are analytically derived. We conducted a simulation study to demonstrate the method's effectiveness, even with small sample sizes. Furthermore, we illustrate the method's application using a published dataset from biological psychology and provide an implementation as an R function.