| Abstract: | Infinitely divisible distributions (i.d.d.'s) with a finite variance have a characteristic function of a particular form. The exponent is written in terms of the canonical or Kolmogorov measure. This paper considers a nonparametric estimate of the Kolmogorov measure based on the empirical characteristic function (e.c.f.) and a truncation. The weak convergence of this estimator is studied. The raw form of the estimator is a functional of the e.c.f., but to be useful in a finite sample it requires some additional smoothing. Thus smoothed estimators are considered. A dynamic data dependent method of truncation is given. A simulation study is undertaken to show how the Kolmogorov measure can be estimated, as well as giving an illustration of the numerical stability questions. It is also seen that a large sample size is needed. |
