1. 
The estimates for the lifetable aging ratio have been smoothed using KalmanFilter
techniques.
 2. 
Madame Jeanne Calment was born on the 21st of February 1875 and died on the 4th of August
1997 at the age of 122 years, 5 months and 14 days. Madame Jeanne Calment age at death is
currently the highest verified age at death of a person.
 3. 
In the period 5th of December 1992 – 31st of December 1993 116, 611 deaths occured in
Bulgaria. A relatively small fraction of 8, 541 deaths (7.33 %), could not be linked to the
census records (4,142 (48.50%) of the unlinked deaths are males, and 4,399 (51.50%) are
females). Although the death certificate in Bulgaria contains some limited information on the
socioeconomic status of dead persons (e.g., education at death, marital status at death, place of
residence at death), we omit the unlinked deaths from our analysis mainly for two reasons: (a) We
do not know whether the persons corresponding to unlinked death certificates have participated
the census, and their personal data were wrongly coded in the census records, or whether they are
subject to census undercount. This latter possibility is supported by the fact that some relevant
minorities in Bulgaria are more likely to be a subject of undercount, which is indicated by the
fact that most of the unlinked deaths involve persons with low education (76.56% of the unlinked
male, and 86.63% of the unlinked female deaths). (b) The unlinked deaths can also correspond
to return migrants whose prevalence has increased after the first waves of rapid emigration in the
early 1990s. We believe that census undercount and return migration are the most important factors
leading to unlinked death certificates. If the persons corresponding to the unlinked deaths have
not participated in the census due to the above reasons, then excluding them from the analyses is
appropriate. If they have participated in the census, but the death cannot be linked due to incomplete
identification numbers, our estimates about the mortality level are biased downward. However, we
have no reasons to believe that this bias is highly sexspecific.
Therefore, the Bulgarian age and sexspecific patterns of mortality presented in this paper
cannot be attributed to a bias caused by unlinked deaths because the overall numbers of unlinked
deaths is quite small and is evenly distributed across sexes. Moreover, the age distribution of the
linked and unlinked deaths is relatively similar for males, while it is slightly shifted to higher ages
for females. For instance, the exact mean age at death of unlinked male deaths is 67.2 years, which
is slightly below that of linked male deaths of 67.8 years, and female unlinked deaths have a mean
age of 76.7, which is above that of linked deaths with a mean age of 73.2.
Age reporting is of relatively high quality in the Bulgarian census. The age of a person is
coded in the personal identification number (PID), which was assigned to the individuals either
during 1976–78 for those alive at this time, or at birth for those born afterwards. The age in the
census form is taken from the age coded in the PID number. Since the old and oldestold in 1992
have received their PID about 15 years earlier, when the propensity to agemisreporting may have
been lower due to younger ages and careful administrative checking, the age information in our
census data should be of high quality even for the old population.
 4. 
The piecewiseconstant proportional hazard model with unobserved frailty is specified as follows.
Consider the ageintervals (c_{0}, c_{1}], ..., (c_{j1}, c_{j}], ..., (c_{K1}, c_{K}] that separate the observed age range into K
disjoint intervals. Then assume that the mortality hazard, conditional on a frailty z = 1 and the
observed covariates y_{a}, is constant within each of these age intervals and equals a(y_{a})m_{j} for
x Î (c_{j1}, c_{j}]. In this specification m_{j} is the mortality hazard prevailing in the age interval (c_{j1}, c_{j}],
j = 1, ..., K, and a(y_{a}) is the factor of proportionality for individuals with characteristics
y_{a}. Denote as s^{PW }(x) the corresponding survival function at age x. The observed hazard
at age x in a heterogeneous population with Gammadistributed relative frailty then equals

(12)

Because of the numerical difficulties in estimating this piecewiseconstant hazard function via maximum
likelihood in the presence of many ageintervals and large data, we implement a slight approximation to
the hazard function (12). In particular, the difficulties in the estimation arise because the hazard
^{PW }(x)
is not constant within age intervals. This results from the fact that the value of the
survival function s^{PW }(x)
in the denominator declines with age x. For sufficiently small ageintervals, however,
the effect of this
changing value of the survival function on the observed hazard
^{PW }
(x) within an age interval is small.
The piecewiseconstant hazard function with relative frailty can therefore be approximated
by replacing the value of the survival function s^{PW }(x) in (12)
with the value of the survival
function at the midpoint of each ageinterval. With this approximation, the observed hazard
^{PW }
(x) is constant within age intervals and the MLE estimation is substantially simplified. We
estimate this piecewiseconstant hazard model with a constant mortality risk within twoyear age
intervals.

