114 V. D’Amato and M. Russolillo
describing the log of a time series of age-specific death ratesmx,tas the sum of
an age-specific parameter independent of timeαx and a component given by the
product of a time-varying parameterκt, reflecting the general level of mortality and
the parameterβx, representing how rapidly or slowly mortality ateach age varies
when the general level of mortality changes. The final termEx,tis the error term,
assumed to be homoschedastic (with mean 0 and varianceσ^2 ).
On the basis of equation( 1 ),ifM ̃x,tis the matrix holding the mean centred
log-mortality rates, the LC model can be expressed as:
M ̃x,t=ln(
Mx,t)
−αx=βxκt+Ex,t. (2)Following LC [7], the parametersβxandκtcan be estimated according to the Sin-
gular Value Decomposition (SVD) with suitable normality constraints. The LC model
incorporates different sources of uncertainty, as discussed in LC [8], Appendix B: un-
certainty in the demographic model and uncertainty in forecasting. The former can
be incorporated by considering errors in fitting the original matrix of mortality rates,
while forecast uncertainty arises from the errors in the forecast of the mortality index.
In our contribution, we deal with the demographic component in order to consider
the sensitivity of the estimated mortality index. In particular, the research consists
in defining an experimental strategy to force the fulfilment of the homoschedasticity
hypothesis and evaluate its impact on the estimatedκt.
3 The experiment
The experimental strategy introduced above, with the aim of inducing the errors to
satisfy the homoschedasticity hypothesis, consists in the following phases [11]. The
error term can be expressed as follows:
Êx,t=M ̃x,t−β̂x̂κt, (3)i.e., as the difference between the matrixM ̃x,t, referring to the mean centred log-
mortality rates and the product betweenβxandκtderiving from the estimation of the
LC model. The successive step consists in exploring the residuals by means of statis-
tical indicators such as: range, interquartile range, mean absolute deviation (MAD) of
a sample of data, standard deviation, box-plot, etc. Afterward, we proceed in finding
those age groups that show higher variability in the errors. Once we have explored
the residualsÊx,t, we may find some non-conforming age groups. We rank them
according to decreasingnon-conformity, i.e., from the more widespread to the more
homogeneous one. For each selected age group, it is possible to reduce the variability
by dividing the entire range into several quantiles, leaving asideeach time the fixed
α% of the extreme values. We replicate each running under the same conditions a
large number of times (i.e., 1000). Foreach age group and for each percentile, we
define a new error matrix. The successive runnings give more and more homogeneous
error terms. By way of this experiment, we investigate the residual’s heteroschedas-
ticity deriving from two factors: the age group effect and the number of altered values