De Moivre's law
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De Moivre's Law is a survival model applied in actuarial science, named for
Abraham de Moivre Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He moved ...
. It is a simple law of mortality based on a linear
survival function The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time. The survival function is also known as the survivor function or reliability function. The term ...
.


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 The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time. The survival function is also known as the survivor function or reliability function. The term ...
: S(x) = 1 - \frac, \qquad 0 \leq x < \omega. In
actuarial notation Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables. Traditional notation uses a halo system where symbols are placed as superscript or subscript before or aft ...
''(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 In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...
that ''(x)'' survives to age ''x+t'' is ''Pr 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 = \frac, \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 Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables. Traditional notation uses a halo system where symbols are placed as superscript or subscript before or aft ...
for conditional probability of failure is _t q_x ''= Pr T(0) ≥ x'. Under de Moivre's law, the probability that ''(x)'' fails to survive to age ''x+t'' is : _t q_x = \frac = \frac. The
force of mortality In actuarial science, force of mortality represents the instantaneous rate of mortality at a certain age measured on an annualized basis. It is identical in concept to failure rate, also called hazard function, in reliability theory. Motivation a ...
(
hazard rate Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysi ...
or
failure rate Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering. The failure rate of a ...
) 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, \qquad 0 \leq t < \omega-x, which has the property of an increasing failure rate with respect to age. 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 table In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, what the probability is that a person of that age will die before their next birthday ("probability of deat ...
s.


History

De Moivre's law first appeared in his 1725 ''Annuities upon Lives'', the earliest known example of an actuarial textbook. Despite the name now given to it, de Moivre himself did not consider his law (he called it a "hypothesis") to be a true description of the pattern of human mortality. Instead, he introduced it as a useful approximation when calculating the cost of annuities. In his text, de Moivre noted that " ... although the Notion of an equable Decrement of Life ...
oes Oes or owes were metallic "O" shaped rings or eyelets sewn on to clothes and furnishing textiles for decorative effect in England and at the Elizabethan and Jacobean court. They were smaller than modern sequins. Making and metals Robert Sharp obta ...
not exactly agree with the ''Tables'', yet that Notion may successfully be employed in constructing a ''Table'' of the Values of ''Annuities'' for ''Ages'' not inferiour to ''Twelve'' ... ". Furthermore, although his text contained an algebraic demonstration that applied to the entire expected future life span, de Moivre also supplied an algebraic demonstration that applied only to a limited number of years. It was this latter result that was used in his subsequent numerical examples. These examples showed de Moivre using his hypothesis in a piecewise fashion, wherein he assumed that the overall pattern of human mortality could be approximated by several straight-line segments (see his illustration to the right). He wrote that "since the Decrements of Life may without any sensible Error be supposed equal, for any short Interval of Time, it follows that if the whole Extent of Life be divided into several shorter Intervals, ... , the Values of ''Annuities'' for Life ... may easily be calculated ... conformably to any ''Table of Observations'', and for any ''Rate of Interest''". For both of the quotes, de Moivre's references to "tables" were to actuarial life tables. Modern authors are not consistent in their treatment of de Moivre's role in the history of mortality laws. On the one hand, Dick London describes de Moivre's law as "the first
continuous probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
to be suggested" for use as a model of human survival. Robert Batten takes a similar view, adding that " e Moivre'shypothesis .. has of course been found unrealistic". In contrast, the surveys of analytical human survival models by Spiegelman and Benjamin do not mention de Moivre at all (in both cases, the surveys start with the work of
Benjamin Gompertz Benjamin Gompertz (5 March 1779 – 14 July 1865) was a British self-educated mathematician and actuary, who became a Fellow of the Royal Society. Gompertz is now best known for his Gompertz law of mortality, a demographic model published in 1 ...
). In his essay on the history of actuarial science, Stephen Haberman does mention de Moivre, but in the section on "Life Insurance Mathematics" and not the one on "Life Tables and Survival Models". A middle ground of sorts was taken by C. W. Jordan in his ''Life Contingencies'', where he included de Moivre in his section on "Some famous laws of mortality", but added that "de Moivre recognized that this was a very rough approximation hose objective wasthe practical one of simplifying the calculation of life annuity values, which in those days was an arduous task". Another indication that de Moivre himself did not consider his "hypothesis" to be a true reflection of human mortality is the fact that he offered two distinct hypotheses in the ''Annuities upon Lives''. When he turned his attention to the question of valuing annuities payable on more than one life, de Moivre found it convenient to drop his assumption of an equal number of deaths (per year) in favor of an assumption of equal probabilities of death at each year of age (i.e., what is now called the "constant
force of mortality In actuarial science, force of mortality represents the instantaneous rate of mortality at a certain age measured on an annualized basis. It is identical in concept to failure rate, also called hazard function, in reliability theory. Motivation a ...
" assumption). Although the constant-force assumption is also recognized today as a simple analytical law of mortality, it has never been known as "de Moivre's second law" or any other such name.De Moivre's introduction of his constant-force assumption starts at page 28 of ''Annuities upon Lives'' and is used extensively through page 49.


Notes

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External links


1725 edition of ''Annuities upon Lives''
Actuarial science Survival analysis