  A continuous-time multi-state history is semi-Markovian, if an intensity to
migrate from one state into another, depends on the duration in the first
state. Such duration can be formalised as covariate, entering the intensity
process of the transition counts. We derive the integrated intensity process,
prove its predictability and the martingale property of the residual. In
particular, we verify the usual conditions for the respective filtration. As a
consequence, according to Nielsen and Linton (1995), a kernel estimator of the
transition intensity, including the duration dependence, converges point-wise
at a slow rate, compared to the Markovian kernel estimator, i.e when ignoring
dependence. By using the rate discrepancy, we follow Gozalo (1993) and show
that the (properly scaled) maximal difference of the two kernel estimators on a
random grid of points is asymptotically chi-square-1-distributed. As a data
example, for a sample of 130,000 German women observed over a period of nine
years, we model the mortality after dementia onset, potentially dependent on
the disease duration. As usual, the models under both hypotheses need to be
enlarged to allow for independent right-censoring. We find a significant effect
of dementia duration, nearly independent of the bandwidth.
