In an early attempt to describe mortality patterns with a mathematical formula, Abraham DeMoivre
[DeMoivre 1725, p. 4, cited in Keyfitz and Smith 1977, p. 273] hypothesized that ‘the number of lives
existing at any age is proportional to the number of years intercepted between the age given and the
extremity of old age’, i.e.,
where w is the maximum attainable age in the population. From this, the hazard rate, or the force of
mortality at age x is defined as
 (4) 
where m^{D}(x) denotes the DeMoivre hazard function. The hazard rate in this example increases towards
infinity as x approaches the maximum attainable age w. The existence of this maximum attainable age
implies that the force of mortality increases faster than in the Gompertz model, especially as x approaches
the maximum age w [for a related discussion see Zelterman 1992].
For the application to contemporary mortality patterns the hazard function m^{D}(x) in (4) is not
sufficiently flexible. We therefore propose a modified DeMoivre hazard given by
 (5) 
where m^{MD}(x) is the modified DeMoivre hazard function. The survival curve s^{MD}(x) corresponding to
the above hazard function is given by
 (6) 
The parameter b in (5) and 6) needs to satisfy b >
in order that the hazard and survival curves are meaningful.
The hazard function m^{MD}(x) defined in (5) has two limiting properties that render it a plausible and
easily interpretable specification. First, the original DeMoivre hazard m^{D}(x) emerges — up to a factor of
proportionality — from the modified DeMoivre hazard m^{MD}(x) when the product bw in (5) approaches
one. This occurs, for instance, when b ®1/w for a fixed maximum attainable age w. Formally this
property is represented as lim _{b®1/w}m^{MD}(x) = lim _{
bw®1}m^{MD}(x) =
m^{D}(x). Second, as the maximum
attainable age w becomes large, the modified DeMoivre hazard approaches the Gompertz hazard
m^{G}(x) = ae^{bx}. That is, the hazard m^{MD}(x) in (5) has the limit lim _{
w®¥}m^{MD}(x) = m^{G}(x). The
reason for this convergence to the Gompertz hazard is easily seen by noting that the limit
lim _{w®¥}(1 
)^{w} = e^{x}.
The interpretation of the parameters in the modified DeMoivre hazard is further facilitated by the fact
that m^{MD}(x)_{x=0} = a and
_{x=0} =
_{x=0} = b, which implies that the modified
DeMoivre hazard agrees with a Gompertz hazard with equal parameters a and b at the age
x = 0.
In heterogeneous populations with unobserved frailty the observed mortality rate differs from m^{MD}(x)
because the population becomes increasingly selected towards lowfrailty individuals with age. If we
assume a Gammadistributed relative frailty model, then the observed hazard rate and survival curve
implied by the modified DeMoivre model, denoted
^{MD}(x) and
^{MD}(x), follow directly from equations (1) and (2) as
 (7) 
and
 (8) 
where m^{MD}(x) and s^{MD}(x) are the hazard function and survival curve in equations (5) and 6) for
individuals with a constant frailty z = 1.
Figure 3 plots the modified DeMoivre hazard function and the implied lifetable aging rate for
different values of s^{2} representing different degrees of unobserved heterogeneity in the population. In this
figure we have used a maximum attainable age w of 122.45 years, i.e., a w that corresponds to Madame
Jeanne Calment age at death, and we used the modified DeMoivre hazard function to model mortality
during ages 40–100. [Note 2] The dasheddotted line in both graphs represents the values for the modified
DeMoivre hazard without unobserved frailty, or equivalently, for individuals with a constant frailty z = 1.
Conditional on a constant frailty z the modified DeMoivre hazard increases faster than the
exponential Gompertz hazard and the lifetable aging rate is also an increasing function of
age. As the age approaches the maximum attainable age w, both the mortality hazard and the
lifetable aging rate approach infinity and the probability of surviving to ages larger than w is
zero.
The existence of such an upper limit to lifespan is in contrast to the Gompertz model that does not
imply a maximum attainable age and it is certainly controversial in view of the recent debate about the
limits to the increase in life expectancy and particularly to the biological limits of lifespan [Gavrilov and
Gavrilova 1991, Manton and Stallard 1996, Vaupel et al. 1998, Wilmoth 1997, Wilmoth et al. 2000]. An
emerging consensus in this debate seems to be that if upper limits to lifespan exist, they need to be seen
in a dynamic perspective [Carey and Judge 2000] and almost certainly do not represent immutable
biological limits that are insurmountable by medical or social progress in survival to very old ages. At the
same time, in any given socioeconomic environment the observed human lifespan is finite and no human
being has been documented to survive above Madame Jeanne Calment’s age at death. Hence,
while there probably does not exist an absolute biological limit to lifespan, human lifespan
in any socioeconomic context may be limited and the changes of this upper limit to life in
itself may constitute an interesting area of research [e.g., Carey and Liedo 1995, Wilmoth and
Lundström 1996].
The mortality hazard in the modified DeMoivre model, conditional on a constant frailty z, increases
faster than exponential and implies an increasing lifetable aging rate with age. Despite this fact, the
observed mortality pattern in a heterogeneous population can reflect a substantially different pattern. In
Figure 3 we have therefore included the observed mortality hazard and lifetable aging rate that is implied
by the modified DeMoivre hazard with different degrees of unobserved heterogeneity in the
population. Most strikingly, the observed mortality pattern does not necessarily represent the
mortality pattern conditional on the frailty z [for an influential related discussion see Vaupel and
Yashin 1985], but reflects properties that are characteristic of the Bulgarian mortality pattern in Figure
1 and also that of other countries. For instance, the observed lifetable aging rate between
ages 40–100 can be initially increasing, and the extent of this increase in the lifetable aging
rate depends on the degree of heterogeneity and the level of mortality. At the same time, the
relative increase in mortality by age decreases at old and oldestold ages. Once age approaches
the maximum attainable lifespan w, this decline in the lifetable aging rate reverses again
and the observed mortality rate and the lifetable aging rate increase and ultimately approach
infinity at w. While this last implication for extremely old ages is controversial in view of the
discussion on the limits to human longevity, the modified DeMoivre hazard in combination
with unobserved heterogeneity can represent observed agepatterns of mortality for the quite
wide agerange from adult to old and oldestold ages during which most deaths in humans
occur.
In the age range below age 100, therefore, the modified DeMoivre hazard function m^{MD}(x) has several
properties that make it a plausible choice for estimating frailty models in mortality. For instance,
nonparametric estimations of the hazard function, which are feasible in bivariate frailty model applied to
twin data [Yashin et al. 1995], have revealed that the hazard for individuals with constant frailty z = 1
during adult ages is increasing substantially faster than the Gompertz hazard, while the observed mortality
rates can be approximated by a Gompertz or Logistic hazard function. In addition, detailed
multivariate followup studies document a differential selection in male and female cohorts with
respect to physiological characteristics and functional abilities that indicate an important role of
selection processes for understanding malefemale differences in mortality [Manton et al. 1995].
On the basis of these and related findings, Caselli et al. [2000, p. 8] have concluded that the
‘correction for unobserved heterogeneity in demographic life tables may be needed not only for
the oldestold but also for the traditional interval of aging between 35 and 85 years of age,
for which the observed trajectory of mortality appears to be welldescribed by a Gompertz
curve.’
The modified DeMoivre hazard function in Figure 3 provides a possibility to estimate frailty models in
the above situation. In particular, the observed hazard
^{MD}(x) as well as the baseline hazard m^{MD}(x)
in this Figure agree highly with the nonparametric estimates reported in Caselli et al. [2000]
and Yashin et al. [1995]. Since nonparametric estimation is only feasible with special data, as
for instance data on the mortality of twins or data with proportionalhazard covariates, the
modified DeMoivre hazard introduced in this paper provides a suitable alternative for frailty
modeling with vital statistics data. The hazard function introduced in this paper, therefore,
allows the investigation of the ‘selectivity hypotheses’ for the convergence of male and female
mortality in Bulgaria, and possibly also other countries, where relativefrailty Gompertz models
with unobserved heterogeneity do not yield an accurate description of the observed mortality
dynamics.
Two basic approaches exist for the estimation of the unknown parameters a, b, w and
s^{2}. First, we can
choose a plausible value for w, such as Madame Jeanne Calment’s age at death. The remaining
parameters a, b, s^{2} can then be estimated via a maximum likelihood using w = 122.45. This
approach should be taken if mortality data at very old ages, say above age 110 that could shed
light on the value of w, are not available or reliable. In our experience, this procedure yields a
quite robust and plausible estimation, and the results of the parameters a and b are not very
sensitive to the choice of w as long as it is chosen within a plausible age range, say 110 – 150
years.
The second possibility is to estimate all four parameters a, b, w and s^{2} using maximum likelihood
estimation. Since w determines the convexity of the hazard function, and s^{2} influences the extent to which
the increase in
(x) is flattened due to the selection process towards lowfrailty individuals,
the joint estimation of all four parameters is not feasible when only one mortality pattern is
observed. It is, however, feasible if mortality patterns of several subpopulations, e.g., by sex or
educational attainment, are analyzed. An example for this estimation is given in Section 3. Quite
naturally, an effective estimation of the lifespan w requires reliable data at very high ages
that may not be available in many countries. If the focus of the investigation is on adult and
oldage mortality, e.g., as in our Bulgarian example on the age range 40–100, an estimation
strategy that assumes a specific value for w and then conducts a sensitivity analysis may be
preferable to the estimation of w from data that are censored at some upper age limit. If w is
nevertheless estimated in these cases, our experience suggests that the resulting ‘bestfitting mortality
curve’ is often characterized by a too low maximum age w, and a too high level of unobserved
heterogeneity.
Independent of which approach regarding the specification of w is chosen, the parameters a, b, and s^{2}
can be functions of characteristics of individuals or subpopulations. In particular, we implement the
estimation of the modified DeMoivre mortality model with
where y_{a}, y_{b}, y_{s} are vectors of covariates that influence the level of the parameters a, b, and s^{2}. For
instance, in the next Section we will estimate a model where some of the parameters can differ by
sex. This dependence of the parameters on sex is incorporated by including a dummy for
females in y_{a}, y_{b} and/or y_{s}. While this specific analysis is a relatively simple dependence of
the parameters a, b, and s^{2} on covariates, considerably more complex specifications are also
feasible.
