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Anastasia Kostaki, Javier M. Moguerza, Alberto Olivares and Stelios Psarakis

Figure 11. Observed and estimated age-specific fertility rates of US, 2003. Figure 12. Observed and estimated age-specific fertility rates of US,
2003. White population. 2003. Black population.

estimators with local bandwidth parameter. And in a similar way, the initial bandwidth parameter is
derived using the KernSmooth library in R package. In particular, for this implementation, an ini-
tial bandwidth of h =1.3118 was obtained.
Also, in the case of mortality and fertility data for the SVM applications, the subroutine svm of the
library e1071 for the R-package was used, and a similar two-step cross-validation technique was
used to select the parameters ε, σ, and C of the ε-regression procedure. In particular, the values ε =
0.00008, σ = 0.0424, and C = 0.0662, have been obtained for this SVM implementation.
Single-year age-specific first-marriage rates for the female populations of Spain, Greece, Italy,
Germany, the Netherlands, Norway, Sweden, Finland, Ireland, and the UK for the available years
were used. These data were obtained from the Eurostat New Cronos database (http://www.eui.eu/
Research/Library/ResearchGuides/ Economics/Statistics/DataPortal/NewCronos.aspx).
The parameters of the parametric models are estimated, as in the case of fertility for the same
reasons mentioned before, by means of a least-squares procedure by minimizing the following sum
of squares.
2
ˆ
∑ ( f − f x ), (4.3)
x
x
ˆ
where f is the estimated first-marriage rate at age x and ƒ x is the empirical one.

x
All the parametric models are fitted by means of a non-linear least-squares procedure and a
Gauss-Newton optimization scheme. The Matlab built-in routine for non-linear parameter estimation
“lsqnonlin” was used in order to find the unconstrained minimum of the unweighted sum of squares.
The residual sums of squares are given in Table 4. Furthermore, the empirical and graduated
age-specific first-marriage rates for selected years and countries are depicted in Figures 13–22.
The observed and estimated rates of GLG, C-M, and SC-M models as well as of SVM are de-
picted in Figures 13–19 for the populations of Germany, Greece, Italy, the Netherlands, Norway,
Spain, and the UK.
The existence of a bulge at young ages and another one at the older ones becomes obvious in re-

International Journal of Population Studies | 2016, Volume 2, Issue 1 13