Force of mortality

In actuarial science and demography, force of mortality, also known as death intensity,^[1][2] is a function, usually written ${\displaystyle \mu (x)}$, that gives the instantaneous rate at which deaths occur at age x, conditional on survival to age x.^[3] In survival analysis it corresponds to the hazard function, and in reliability theory it corresponds to the failure rate.^[4][5] It has units of inverse time, and integrating it over an interval gives the survival probability over that interval.^[4]

Let ${\displaystyle X}$ be a non-negative random variable representing an individual's age at death (or lifetime). Write ${\displaystyle F(x)=\mathrm{Pr}(X\le x)}$ for its cumulative distribution function and ${\displaystyle S(x)=\mathrm{Pr}(X>x)}$ for its survival function.^[4]

The force of mortality at age ${\displaystyle x}$, written ${\displaystyle \mu (x)}$, is defined as the instantaneous conditional rate of death at age ${\displaystyle x}$. Formally, it is the limit of the conditional probability of dying in a short interval after ${\displaystyle x}$, divided by the interval length^[3]: $${\displaystyle \mu (x)=\underset{\mathrm{\Delta }x\to 0^{+}}{\lim }\frac{\mathrm{Pr}(x<X\le x+\mathrm{\Delta }x\mid X>x)}{\mathrm{\Delta }x}.}$$

When ${\displaystyle X}$ is continuous with probability density function ${\displaystyle f(x)}$, the force of mortality can be written in terms of ${\displaystyle f}$ and ${\displaystyle S}$ as^[4] $${\displaystyle \mu (x)=\frac{f(x)}{S(x)}=\frac{f(x)}{1-F(x)}.}$$

Equivalently, where ${\displaystyle S}$ is differentiable, it is the negative derivative of the log-survival function^[4]: $${\displaystyle \mu (x)=-\frac{\mathrm{d}}{\mathrm{d}x}\ln S(x).}$$

The force of mortality ${\displaystyle \mu (x)}$ is an instantaneous rate rather than a probability. For a short interval ${\displaystyle \mathrm{\Delta }x}$, the conditional probability of dying shortly after age ${\displaystyle x}$ is approximately ${\displaystyle \mu (x)\mathrm{\Delta }x}$, provided ${\displaystyle \mathrm{\Delta }x}$ is small enough that the rate does not change much over the interval.^[3]

In survival analysis, ${\displaystyle \mu (x)}$ is the hazard function.^[4] In reliability theory, the same mathematical object is commonly called the failure rate.^[5]

The cumulative force of mortality (also called the cumulative hazard) is the integral of the force over age. Writing $${\displaystyle H(x)={\displaystyle \underset{0}{\overset{x}{\int }}}\mu (u)\mathrm{d}u}$$ then the survival function can be expressed as^[4] $${\displaystyle S(x)=\exp (-H(x)).}$$

These identities imply the differential relationship $${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}S(x)=-\mu (x)S(x)}$$ and, for a continuous lifetime distribution, the density can be written as^[4] $${\displaystyle f(x)=\mu (x)S(x).}$$

In actuarial notation, the probability that a life aged ${\displaystyle x}$ survives for a further ${\displaystyle t}$ years is written ${\displaystyle {}_t{p}_x}$. In terms of the lifetime random variable ${\displaystyle X}$, it is^[3] $${\displaystyle {}_t{p}_x=\mathrm{Pr}(X>x+t\mid X>x)=\frac{S(x+t)}{S(x)}.}$$

Using the force of mortality, this conditional survival probability can be expressed as an exponential of the integrated force^[3][4]: $${\displaystyle {}_t{p}_x=\exp (-{\displaystyle \underset{x}{\overset{x+t}{\int }}}\mu (u)\mathrm{d}u).}$$

Life tables often tabulate survival and death probabilities at integer ages. In that setting, the one-year survival probability is ${\displaystyle p_x={}_1{p}_x}$ and the one-year death probability is ${\displaystyle q_x=1-p_x}$.^[3] The force of mortality provides a continuous-age description that can be used to relate probabilities over different intervals through the integral relationship above.^[3]

Several parametric models are used to describe how the force of mortality varies with age. A constant force of mortality, ${\displaystyle \mu (x)=\lambda }$ for ${\displaystyle \lambda >0}$, corresponds to an exponential distribution for ${\displaystyle X}$ and gives a memoryless survival pattern.^[4]

In actuarial work, the Gompertz–Makeham law of mortality is often written as the sum of an age-independent component and an exponentially increasing component, for example $${\displaystyle \mu (x)=A+B{c}^x}$$ with ${\displaystyle A\ge 0}$, ${\displaystyle B>0}$, and ${\displaystyle c>1}$.^[3] The Gompertz model is the special case ${\displaystyle A=0}$, giving:^[3] $${\displaystyle \mu (x)=B{c}^x}$$

A common model in survival analysis and reliability uses a Weibull hazard, which has the form $${\displaystyle \mu (x)=\frac{k}{\lambda }{\left(\frac{x}{\lambda }\right)}^{k-1}}$$ for shape ${\displaystyle k>0}$ and scale ${\displaystyle \lambda >0}$. This family includes decreasing, constant, and increasing forces of mortality depending on the value of ${\displaystyle k}$.^[4][5]

• Failure rate
• Hazard function
• Actuarial present value
• Actuarial science
• Reliability theory
• Life expectancy

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