Tests are given for the Laplace or double exponential distribution. The test statistics are based on the empirical distribution function and include the families of Cramér-von Mises and Kolmogorov-Smirnov. Asymptotic theory is given, and asymptotic points are calculated, for the Cramér-von Mises family, and Monte Carlo points for finite samples are given for all the statistics. Power studies suggest that the Watson statistic is the most powerful for the common problem of testing Laplace against other symmetric distributions. An application of the Laplace distribution is in LAD (or L1) regression. This is also discussed in the article, with two examples.
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Puig et al. (2000) studied this question.
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