Parametric Mortality Models

hazard(model,age)

The force of mortality at age: the density of the age at death divided by the probability of surviving to age, $\mu_x = f(x)/S(x) = -S'(x)/S(x)$.

Every parametric law implements hazard; cumhazard, survival, and decrement follow from it.

cumhazard(model,age)
cumhazard(model,from_age,to_age)

The integrated hazard up to age, $H(x) = \int_0^x \mu_s \, ds$, equivalently $-\log S(x)$, so that survival(model, age) == exp(-cumhazard(model, age)). The two-age form integrates from from_age to to_age: $-\log(S(\text{to})/S(\text{from}))$.

The generic method integrates hazard numerically from age zero. A law may override the one-argument form with a closed-form expression; the two-age form is the difference of the one-argument form, so a closed form is used automatically wherever it exists.

μ(model,age)

$\mu_x$: the force of mortality at age. An alias for hazard.

Makeham(;a,b,c)

Construct a mortality model following Makeham's law.

$\mathrm{hazard} \left( {\rm age} \right) = ae^{bx} + c$

Default args:

a = 0.0002
b = 0.13
c = 0.001

Gompertz(;a,b)

Construct a mortality model following Gompertz' law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = ae^{bx}$

This is a special case of Makeham's law and will Makeham model where c=0.

Default args:

a = 0.0002
b = 0.13

Makeham2(; μ=49, σ=7.692308, c=0.001)

Makeham's law in location/scale parameters, with force of mortality $h(x) = e^{(x-μ)/σ}/σ + c$. It is equivalent to Makeham(a=exp(-μ/σ)/σ, b=1/σ, c=c).

μ must be finite, σ finite and positive, and c finite and nonnegative. mu and sigma are ASCII aliases for μ and σ; passing both spellings of a keyword is an ArgumentError. Ages must be nonnegative.

The location and scale are retained as fields: transforming them to a and b can underflow a even when mortality near age μ is representable.

Gompertz2(; μ=49, σ=7.7)

Gompertz's law in location/scale parameters, with force of mortality $h(x) = e^{(x-μ)/σ}/σ$. Returns a Makeham2 with c=0, equivalent to Gompertz(a=exp(-μ/σ)/σ, b=1/σ). mu and sigma are ASCII keyword aliases.

Carriere1(; p₁=0.003, p₂=0.007, μ₁=2.7, μ₂=3, μ₃=88, σ₁=15, σ₂=6, σ₃=9.5)

Carrière's three-component survival mixture: Weibull, inverse Weibull, and Gompertz. For weights $w = (p₁, p₂, 1-p₁-p₂)$ and component survival functions $S_i$,

\[S(x) = \sum_{i=1}^3 w_i S_i(x), \qquad H(x) = -\log S(x), \qquad h(x) = \frac{\sum_{i=1}^3 w_i S_i(x) h_i(x)}{S(x)}.\]

The components are Weibull(m=μ₁, σ=σ₁), InverseWeibull(m=μ₂, σ=σ₂), and Gompertz2(μ=μ₃, σ=σ₃). Each is normalized to survival one at age zero. The hazard is an instantaneous force, not an annual mortality probability; use decrement(model, age, age + 1) for the latter.

Weights must be nonnegative with p₁ + p₂ ≤ 1; zero weights are allowed. All locations and scales must be finite and positive. Ages must be nonnegative. The hazard at zero is infinite when p₁ > 0 and μ₁ < σ₁ (an integrable Weibull singularity); survival there is one. The inverse-Weibull tail gives an infinite mean lifetime when p₂ > 0 and μ₂/σ₂ ≤ 1, as with the defaults. The defaults are inherited from MortalityLaws, not a calibrated mortality table: their hazard declines sharply at old ages as inverse-Weibull survivors dominate.

This implementation uses σ₂ for the inverse-Weibull component, following Carrière's paper. Older MortalityLaws implementations used σ₁ there and ignored σ₂; that wiring was corrected in MortalityLaws' v2.3.0 development commit 8f13abb. Parameters fitted with the older formula generally need refitting before use here. MortalityLaws also reports finite differences of cumulative hazard, whereas hazard here is the instantaneous force.

The zero-age limits above apply to ordinary numeric ages. ForwardDiff age derivatives at exactly zero are not reliable for InverseWeibull or the Carrière mixtures. They can return NaN where the right-hand derivative is finite, or where it is +Inf because of a singular Weibull component (as in both mixtures' defaults, including Carriere2, which has no inverse-Weibull component).

ASCII aliases are p1, p2, mu1 to mu3, and sigma1 to sigma3. Passing both spellings of a keyword is an ArgumentError.

See Carrière (1992), Parametric Models for Life Tables, Transactions of the Society of Actuaries 44, 77–99, and the MortalityLaws implementation.

Carriere2(; p₁=0.01, p₂=0.01, μ₁=1, μ₂=49, μ₃=49, σ₁=2, σ₂=7, σ₃=7)

Carrière's survival mixture of Weibull(m=μ₁, σ=σ₁), InverseGompertz(m=μ₂, σ=σ₂), and Gompertz2(μ=μ₃, σ=σ₃). With $w = (p₁,p₂,1-p₁-p₂)$, $S = ∑ᵢ wᵢ Sᵢ$, $H = -log(S)$, and $h = (∑ᵢ wᵢ Sᵢ hᵢ)/S$. Each component is normalized at age zero.

Parameter validation, ASCII keyword aliases, and the hazard's limit at age zero are as for Carriere1. Unlike Carriere1, this mixture has a finite mean lifetime for every allowed parameter set.

InverseGompertz(;m,σ)

Construct a mortality model following InverseGompertz's law.

\[\begin{aligned} \mathrm{hazard} \left( {\rm age} \right) &= \frac{1}{\sigma} \cdot \frac{e^{\frac{ - \left( {\rm age} - m \right)}{\sigma}}}{e^{e^{\frac{ - \left( {\rm age} - m \right)}{\sigma}}} - 1} \\ \mathrm{survival} \left( {\rm age} \right) &= \frac{1 - e^{ - e^{\frac{ - \left( {\rm age} - m \right)}{\sigma}}}}{1 - e^{ - e^{\frac{m}{\sigma}}}} \end{aligned}\]

sigma is an ASCII alias for σ.

Default args:

m = 49
σ = 7.7

Opperman(;a,b,c)

Construct a mortality model following Opperman's law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = \max\left( \frac{a}{\sqrt{{\rm age} + 1}} - b + c \cdot \sqrt{{\rm age} + 1}, 0 \right)$

Default args:

a = 0.04
b = 0.0004
c = 0.001

Thiele(;a,b,c,d,e,f,g)

Construct a mortality model following Thiele's law of mortality.

\[\begin{aligned} \mu_1 &= a \cdot \exp\left( -b \cdot {\rm age} \right) \\ \mu_2 &= c \cdot \exp\left( -\frac{d}{2} \left( {\rm age} - e \right)^{2} \right) \\ \mu_3 &= f \cdot \exp\left( g \cdot {\rm age} \right) \\ \mathrm{hazard} \left( {\rm age} \right) &= \begin{cases} \mu_1 + \mu_3 & \text{if } \left( {\rm age} = 0 \right)\\ \mu_1 + \mu_2 + \mu_3 & \text{otherwise} \end{cases} \end{aligned}\]

Default args:

a = 0.02474 
b = 0.3
c = 0.004
d = 0.5
e = 25
f = 0.0001
g = 0.13

Wittstein(;a,b,m,n)

Construct a mortality model following Wittstein's law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = \frac{1}{b} \cdot a^{ - \left( b \cdot {\rm age} \right)^{n}} + a^{ - \left( m - {\rm age} \right)^{n}}$

Default args:

a = 1.5
b = 1.
n = 0.5
m = 100

The law is defined up to age m, which omega returns. An age past m is a DomainError.

Weibull(;m,σ)

Construct a mortality model following Weibull's law of mortality.

Note that if σ > m, then the mode of the density is 0 and hx is a non-increasing function of x, while if σ < m, then the mode is greater than 0 and hx is an increasing function.

• m >0 is a measure of location
• σ >0 is measure of dispersion

\[\begin{aligned} \mathrm{hazard} \left( {\rm age} \right) = \frac{1}{\sigma} \cdot \left( \frac{{\rm age}}{m} \right)^{\frac{m}{\sigma} - 1} \\ \mathrm{cumhazard} \left( {\rm age} \right) = \left( \frac{{\rm age}}{m} \right)^{\frac{m}{\sigma}} \\ \mathrm{survival} \left( {\rm age} \right) = e^{ - \mathrm{cumhazard} \left( {\rm age} \right)} \end{aligned}\]

sigma is an ASCII alias for σ.

Default args:

m = 1
σ = 2

InverseWeibull(;m,σ)

Construct a mortality model following the Inverse-Weibull law of mortality.

The Inverse-Weibull proves useful for modelling the childhood and teenage years, because the logarithm of h(x) is a concave function.

• m >0 is a measure of location
• σ >0 is measure of dispersion

\[\begin{aligned} \mathrm{hazard} \left( {\rm age} \right) &= \frac{\frac{1}{\sigma} \cdot \left( \frac{{\rm age}}{m} \right)^{\frac{ - m}{\sigma} - 1}}{e^{\left( \frac{{\rm age}}{m} \right)^{\frac{ - m}{\sigma}}} - 1} \\ \mathrm{cumhazard}\left( {\rm age} \right) &= - \log\left( 1 - e^{ - \left( \frac{{\rm age}}{m} \right)^{\frac{ - m}{\sigma}}} \right) \\ \mathrm{survival}\left( {\rm age} \right) &= e^{ - \mathrm{cumhazard}\left( {\rm age} \right)} \end{aligned}\]

sigma is an ASCII alias for σ.

Default args:

m = 5
σ = 10

Perks(;a,b,c,d)

Construct a mortality model following Perks' law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = \frac{a + b \cdot c^{{\rm age}}}{b \cdot c^{ - {\rm age}} + 1 + d \cdot c^{{\rm age}}}$

Default args:

a = 0.002
b = 0.13
c = 0.01
d = 0.01

VanderMaen(;a,b,c,i,n)

Construct a mortality model following VanderMaen's law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = a + b \cdot {\rm age} + c \cdot {\rm age}^{2} + \frac{i}{n - {\rm age}}$

Default args:

a = 0.01
b = 1
c = 0.01
i = 100
n = 200

The law is defined up to age n, where the hazard has a pole; omega returns n. An age past n is a DomainError.

VanderMaen2(;a,b,i,n)

Construct a mortality model following VanderMaen2's law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = a + b \cdot {\rm age} + \frac{i}{n - {\rm age}}$

Default args:

a = 0.01
b = 1
i = 100
n = 200

The law is defined up to age n, where the hazard has a pole; omega returns n. An age past n is a DomainError.

StrehlerMildvan(;k,v₀,b,d)

Construct a mortality model following StrehlerMildvan's law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = k \cdot e^{\frac{\left( - v_0 \right) \cdot \left( 1 - b \cdot {\rm age} \right)}{d}}$

v0 is an ASCII alias for v₀.

Default args:

k   = 0.01
v₀  = 2.5
b   = 0.2
d   = 6.0

Beard(;a,b,k)

Construct a mortality model following Beard's law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = \frac{a \cdot e^{b \cdot {\rm age}}}{1 + k \cdot a \cdot e^{b \cdot {\rm age}}}$

Default args:

a = 0.002
b = 0.13
k = 1.

MakehamBeard(;a,b,c,k)

Construct a mortality model following MakehamBeard's law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = \frac{a \cdot e^{b \cdot {\rm age}}}{1 + k \cdot a \cdot e^{b \cdot {\rm age}}} + c$

Default args:

a = 0.002
b = 0.13
c = 0.01
k = 1.

Quadratic(;a,b,c)

Construct a mortality model following Quadratic law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = a + b \cdot {\rm age} + c \cdot {\rm age}^{2}$

Default args:

a = 0.01
b = 1.
c = 0.01

GammaGompertz(;a,b,γ)

Construct a mortality model following GammaGompertz law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = \frac{a \cdot e^{b \cdot {\rm age}}}{1 + \frac{a \cdot \gamma}{b} \cdot \left( e^{b \cdot {\rm age}} - 1 \right)}$

gamma is an ASCII alias for γ.

Default args:

a = 0.002
b = 0.13
γ = 1

Siler(;a,b,c,d,e)

Construct a mortality model following Siler law of mortality.

$\mathrm{hazard} \left( {\rm age} \right) = a \cdot \exp\left( -b \cdot {\rm age} \right) + c + d \cdot \exp\left( e \cdot {\rm age} \right)$

Default args:

a = 0.0002
b = 0.13
c = 0.001
d = 0.001
e = 0.013

HeligmanPollard(;a,b,c,d,e,f,g,h)

Construct a mortality model following HeligmanPollard law of mortality with 8 parameters.

\[\begin{aligned} \mu_1 &= a^{\left( {\rm age} + b \right)^{c}} + g \cdot h^{{\rm age}} \\ \mu_2 &= d \cdot \exp\left( -e \cdot \left( \log\left( \frac{{\rm age}}{f} \right) \right)^{2} \right) \\ \eta &= \begin{cases} \mu_1 & \text{if } \left( {\rm age} = 0 \right)\\ \mu_1 + \mu_2 & \text{otherwise} \end{cases} \\ \mathrm{hazard}\left( {\rm age} \right) &= \frac{\eta}{1 + \eta} \end{aligned}\]

Default args:

a = .0005
b = .004
c = .08
d = .001
e = 10
f = 17
g = .00005
h = 1.1

HeligmanPollard2(;a,b,c,d,e,f,g,h)

Construct a mortality model following HeligmanPollard (alternate) law of mortality with 8 parameters.

\[\begin{aligned} \mu_1 &= a^{\left( {\rm age} + b \right)^{c}} + \frac{g \cdot h^{{\rm age}}}{1 + g \cdot h^{{\rm age}}} \\ \mu_2 &= d \cdot \exp\left( -e \cdot \left( \log\left( \frac{{\rm age}}{f} \right) \right)^{2} \right) \\ \mathrm{hazard}\left( {\rm age} \right) &= \begin{cases} \mu_1 & \text{if } \left( {\rm age} = 0 \right)\\ \mu_1 + \mu_2 & \text{otherwise} \end{cases} \end{aligned}\]

Default args:

a = .0005
b = .004
c = .08
d = .001
e = 10
f = 17
g = .00005
h = 1.1

HeligmanPollard3(;a,b,c,d,e,f,g,h,k)

Construct a mortality model following HeligmanPollard (alternate) law of mortality with 9 parameters.

\[\begin{aligned} \mu_1 &= a^{\left( {\rm age} + b \right)^{c}} + \frac{g \cdot h^{{\rm age}}}{1 + k \cdot g \cdot h^{{\rm age}}} \\ \mu_2 &= d \cdot \exp\left( -e \cdot \left( \log\left( \frac{{\rm age}}{f} \right) \right)^{2} \right) \\ \mathrm{hazard}\left( {\rm age} \right) &= \begin{cases} \mu_1 & \text{if } \left( {\rm age} = 0 \right)\\ \mu_1 + \mu_2 & \text{otherwise} \end{cases} \end{aligned}\]

Default args:

a = .0005
b = .004
c = .08
d = .001
e = 10
f = 17
g = .00005
h = 1.1
k= 1.

HeligmanPollard4(;a,b,c,d,e,f,g,h,k)

Construct a mortality model following HeligmanPollard (alternate) law of mortality with 9 parameters.

\[\begin{aligned} \mu_1 &= a^{\left( {\rm age} + b \right)^{c}} + \frac{g \cdot h^{{\rm age}^{k}}}{1 + g \cdot h^{{\rm age}^{k}}} \\ \mu_2 &= d \cdot \exp\left( -e \cdot \left( \log\left( \frac{{\rm age}}{f} \right) \right)^{2} \right) \\ \mathrm{hazard}\left( {\rm age} \right) &= \begin{cases} \mu_1 & \text{if } \left( {\rm age} = 0 \right)\\ \mu_1 + \mu_2 & \text{otherwise} \end{cases} \end{aligned}\]

Default args:

a = .0005
b = .004
c = .08
d = .001
e = 10
f = 17
g = .00005
h = 1.1
k= 1.

RogersPlanck(;a₀, a₁, a₂, a₃, a, b, c, d, u)

Construct a mortality model following RogersPlanck law of mortality.

$\mathrm{hazard}\left( {\rm age} \right) = a_0 + a_1 \cdot e^{\left( - a \right) \cdot {\rm age}} + a_2 \cdot e^{b \cdot \left( {\rm age} - u \right) - e^{\left( - c \right) \cdot \left( {\rm age} - u \right)}} + a_3 \cdot e^{d \cdot {\rm age}}$

a0, a1, a2 and a3 are ASCII aliases for a₀, a₁, a₂ and a₃.

Default args:

a₀ = 0.0001
a₁ = 0.02
a₂ = 0.001
a₃ = 0.0001
a  = 2.
b  = 0.001
c  = 100.
d  = 0.1
u  = 0.33

Martinelle(;a,b,c,d,k)

Construct a mortality model following Martinelle's law of mortality.

$\mathrm{hazard}\left( {\rm age} \right) = \frac{a \cdot e^{b \cdot {\rm age}} + c}{1 + d \cdot e^{b \cdot {\rm age}}} + k \cdot e^{b \cdot {\rm age}}$

Default args:

a = 0.001
b = 0.13
c = 0.001
d = 0.1
k = 0.001

Kostaki(;a,b,c,d,e1,e2,f,g,h)

Construct a mortality model following Kostaki's law of mortality. A nine-parameter adaptation of HeligmanPollard.

\[\begin{aligned} \mu_1 &= a^{\left( {\rm age} + b \right)^{c}} + g \cdot h^{{\rm age}} \\ \mu_2 &= \begin{cases} d \cdot e^{ - \left( e1 \cdot \log\left( \frac{{\rm age}}{f} \right) \right)^{2}} & \text{if } \left( {\rm age} \leq f \right)\\ d \cdot e^{ - \left( e2 \cdot \log\left( \frac{{\rm age}}{f} \right) \right)^{2}} & \text{otherwise} \end{cases} \\ \eta &= \begin{cases} \mu_1 & \text{if } \left( {\rm age} = 0 \right)\\ \mu_1 + \mu_2 & \text{otherwise} \end{cases} \\ \mathrm{hazard}\left( {\rm age} \right) &= \frac{\eta}{1 + \eta} \end{aligned}\]

Default args:

a = 0.0005
b = 0.01
c = 0.10
d = 0.001
e1 = 3.
e2 = 0.1
f = 25.
g = .00005
h = 1.1

Kostaki, A. (1992). A nine‐parameter version of the Heligman‐Pollard formula. Mathematical Population Studies, 3(4), 277–288. doi:10.1080/08898489209525346

Kannisto(;a,b)

Construct a mortality model following Kannisto's law of mortality.

\[\begin{aligned} \mathrm{hazard}\left( {\rm age} \right) &= \frac{a \cdot e^{b \cdot {\rm age}}}{1 + a \cdot e^{b \cdot {\rm age}}} \\ \mathrm{cumhazard}\left( {\rm age} \right) &= \frac{1}{b} \log\left( \frac{1 + a \cdot e^{b \cdot {\rm age}}}{1 + a} \right) \\ \mathrm{survival}\left( {\rm age} \right) &= e^{ - \mathrm{cumhazard}\left( m, {\rm age} \right)} \end{aligned}\]

Default args:

a = 0.5
b = 0.13

KannistoMakeham(;a,b,c)

Construct a mortality model following KannistoMakeham's law of mortality.

$\mathrm{hazard}\left( {\rm age} \right) = \frac{a \cdot e^{b \cdot {\rm age}}}{1 + a \cdot e^{b \cdot {\rm age}}} + c$

Default args:

a = 0.5
b = 0.13
c = 0.001