Goodness-of-fit Tests I wish to test whether a large number of observations $X_i$ follows an exponential distribution with parameter $\lambda=1$. I also wish to test this hypothesis exactly, and intend that if the observations follow an exponential distribution with a different parameter, the test should reject the null hypothesis given sufficiently many observations. In addition, I want to have a numeric statistic that I could report and do not want the procedure to involve rounding off observation numbers into bins.
Which of these goodness-of-fit tests would be the most appropriate for this purpose?

*

*Chi-squared Test

*Kolmogorov-Smirnov Test

*Kolmogorov-Liliefors Test

*Quantile-quantile plots

I'm opting for Kolmogorov-Smirnov Test. Is this the best one?
I could also go for QQ-plots but it's a graphical method and I want to test the hypothesis exactly. Maybe someone with experience on this field could share their thoughts.
 A: If you want to test the hypothesis of the observations being i.i.d. draws from an exponential(1) distribution, yes, you could use a Kolmogorov-Smirnov. Of the tests you mention, a chi-squared test of an exponential(1) is possible (but involves several more choices, such as the number of bins and the locations of the bin boundaries).
Typically, of those two, the Kolmogorov-Smirnov would tend to have better power against a wide range of alternatives of typical interest, but there are many other tests of fully specified distribution (including Cramer-von Mises and Anderson-Darling). Which you might choose depends what sorts of alternatives you seek power against. The Anderson Darling would tend to do better than the Kolmogorov-Smirnov against heavier-tailed alternatives, for example but would often be less effective at picking up a shift of probability in the middle of the distribution.
None of these could be claimed to be "the most" appropriate without an explicit definition of appropriateness. They all have advantages and disadvantages. Like many omnibus tests, all of the tests mentioned here are biased against some alternatives, at least at some sample sizes.
The Lilliefors test for an exponential would test for an exponential with an unspecified parameter. The chi-squared test could be used here as well with the loss of one d.f. for the parameter estimation (if that was based on the binned data; if the estimate was based on unbinned data, the situation is more complicated). There are many other tests for general exponentiality.
