Bounded type (mathematics)

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De Moivre's Law is a survival model applied in actuarial science, named for Abraham de Moivre. It is a simple law of mortality based on a linear survival function.

Definition

De Moivre's law has a single parameter $\omega$ called the ultimate age. Under de Moivre's law, a newborn has probability of surviving at least x years given by the survival function

$S(x)=1-{\frac {x}{\omega }},\qquad 0\leq x<\omega .$

In actuarial notation (x) denotes a status or life that has survived to age x, and T(x) is the future lifetime of (x) (T(x) is a random variable). The conditional probability that (x) survives to age x+t is Pr[T(0) ≥ x+t | T(0) ≥ x] = S(x+t) / S(x), which is denoted by ${}_{t}p_{x}$. Under de Moivre's law, the conditional probability that a life aged x years survives at least t more years is

${}_{t}p_{x}={\frac {S(x+t)}{S(x)}}={\frac {\omega -(x+t)}{\omega -x}},\qquad 0\leq t<\omega -x,$

and the future lifetime random variable T(x) therefore follows a uniform distribution on $(0,\,\omega -x)$.

The actuarial notation for conditional probability of failure is ${}_{t}q_{x}$ = Pr[0 ≤ T(x) ≤ t|T(0) ≥ x]. Under de Moivre's law, the probability that (x) fails to survive to age x+t is

${}_{t}q_{x}={\frac {S(x)-S(x+t)}{S(x)}}={\frac {t}{\omega -x}}.$

The force of mortality (hazard rate or failure rate) is $\mu (x)=-S'(x)/S(x)=f(x)/S(x),$ where f(x) is the probability density function. Under de Moivre's law, the force of mortality for a life aged x is

$\mu (x+t)={\frac {1}{\omega -(x+t)}},\qquad 0\leq t<\omega -x,$

which has the property of increasing failure rate (IFR) with respect to age that is usually assumed for humans, or anything subject to aging.

De Moivre's law is applied as a simple analytical law of mortality and the linear assumption is also applied as a model for interpolation for discrete survival models such as life tables.

Linear assumption for fractional years

When applied for interpolation, the linear assumption is called uniform distribution of death (UDD) assumption in fractional years and it is equivalent to linear interpolation. If $\ell _{x}$ denotes the number of survivors at exact age x years out of an initial cohort of $\ell _{0}$ lives, the UDD assumption for fractional years is that

$\ell _{x+t}=(1-t)\ell _{x}+t\ell _{x+1},\qquad 0<t<1,$

or equivalently, that

$S(x+t)=(1-t)S(x)+tS(x+1),\qquad 0<t<1.$

Under the UDD assumption, the probability ${}_{t}q_{x}$ that a life aged x fails within (0,t), is $tq_{x}$, and $\mu (x+t)={\frac {q_{x}}{1-tq_{x}}}$, for $\,0<t<1$.

Notes

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