Fig. 1

Repartition of the p-values in a K.S. test of exponentiality with plug-in. Graph of the p-values as a function of their rank. A $n=40$ i.i.d. sample of exponential variables with parameter $\lambda =20$ has been drawn 1000 times. Each time a p-value has been computed either by estimating the parameter λ and performing the K.S. test with exactly the same sample (in black), or by estimating the parameter λ on half of the sample and performing the K.S. test on the other half (in red), or by estimating the parameter λ on the whole sample and performing the K.S. test on a subsample of size $n^{2/3}$ (in green). Note that the estimated level (i.e., the number of p-values smaller than 0.05 divided by 1000) is in the first case of 0.009, of 0.12 in the second case, and of 0.039 in the third case. Those levels and curves are stable with respect to the sample size: Similar results are obtained for a larger sample size ($n=200$ and $n=1000$)