Full text: Applications of event history analysis in life course research

663 
5. Is it possible to evaluate the renewal function numerically? 
The difficulty of evaluating the renewal function numerically is 
tuofold: both the evaluation of the Laplace transform (F,) and the 
inversion of the Laplace transform to evaluate m(t) are numerically 
sensitive operations. 
The numerical evaluation of the Laplace transform given in (15) is 
very difficult. Since the Laplace transform parameter (1) is a complex 
number, its sine and cosine components result in very large fluctuations 
when t is close to zero. Figure 3 presents the L(t) function whose integral 
is the Laplace transform of the Erlangian density with shape parameter 2. 
It can be seen that both the real and the imaginary parts of L(t) fluctuate 
when t is small. However, nearly all numerical integration algorithms 
assume reasonable smoothness of the function to be integrated. When this 
assumption does not hold, the F values become very sensitive to the choice 
of the parameters of the numerical algorithm. 
The second source of difficulty lies in the numerical inversion of the 
Laplace transform of the renewal function. The numerical inversion of the 
Laplace transform is a difficult problem known to many branches of the 
engineering sciences (for an overview of such problems see Bellman et al., 
1966). This is mainly because of the instability of the Laplace 
transformation: a small perturbation in the original function may have a 
large impact on its Laplace transform. For a reasonably accurate numerical 
inversion, the Laplace transform itself has to be evaluated to a 
considerable precision. This is not possible unless one uses extra¬ 
ordinarily accurate numerical algorithms of integration.
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.

powered by Goobi viewer