  The estimation of mutation probabilities and relative fitnesses in
fluctuation analysis is based on the unrealistic hypothesis that the
single-cell times to division are exponentially distributed. Using the
classical Luria-Delbr\"{u}ck distribution outside its modelling hypotheses
induces an important bias on the estimation of the relative fitness. The model
is extended here to any division time distribution. Mutant counts follow a
generalization of the Luria-Delbr\"{u}ck distribution, which depends on the
mean number of mutations, the relative fitness of normal cells compared to
mutants, and the division time distribution of mutant cells. Empirical
probability generating function techniques yield precise estimates both of the
mean number of mutations and the relative fitness of normal cells compared to
mutants. In the case where no information is available on the division time
distribution, it is shown that the estimation procedure using constant division
times yields more reliable results. Numerical results both on observed and
simulated data are reported.
