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〈U〉《the ~》《集合的に》(法律・法規を総称して)『法』 / 〈U〉〈C〉(個々の)『法律』,法規 / 〈U〉法の[統制]力 / 〈U〉法律学 / 〈U〉弁護士[業] / 〈U〉《the law》警察[力],警官(police) / 〈U〉法の適用(発動);訴訟(legal action) / 〈C〉(科学・芸術などでの)法則,きまり / 〈C〉〈U〉(一般に従うべき)おきて,ならわし,規則
ラ(全音階の第6音)
liter[s]

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/07/30 09:30:25」(JST)

wiki en

The Gompertz–Makeham law states that the human death rate is the sum of an age-independent component (the Makeham term, named after William Makeham)^[1] and an age-dependent component (the Gompertz function, named after Benjamin Gompertz),^[2] which increases exponentially with age.^[3] In a protected environment where external causes of death are rare (laboratory conditions, low mortality countries, etc.), the age-independent mortality component is often negligible. In this case the formula simplifies to a Gompertz law of mortality. In 1825, Benjamin Gompertz proposed an exponential increase in death rates with age.

The Gompertz–Makeham law of mortality describes the age dynamics of human mortality rather accurately in the age window from about 30 to 80 years of age. At more advanced ages, some studies have found that death rates increase more slowly – a phenomenon known as the late-life mortality deceleration^[3] – but more recent studies disagree.^[4]

Estimated probability of a person dying at each age, for the U.S. in 2003 [1]. Mortality rates increase exponentially with age after age 30.

The decline in the human mortality rate before the 1950s was mostly due to a decrease in the age-independent (Makeham) mortality component, while the age-dependent (Gompertz) mortality component was surprisingly stable.^[3]^[5] Since the 1950s, a new mortality trend has started in the form of an unexpected decline in mortality rates at advanced ages and "rectangularization" of the survival curve.^[6]^[7]

The hazard function for the Gompertz-Makeham distribution is most often characterised as $h(x)=\alpha e^{\beta x}+\lambda$ . The empirical magnitude of the beta-parameter is about .085, implying a doubling of mortality every .69/.085 = 8 years (Denmark, 2006).

The quantile function can be expressed in a closed-form expressions using the Lambert W function:^[8]

The Gompertz law is the same as a Fisher–Tippett distribution for the negative of age, restricted to negative values for the random variable (positive values for age).

References

^ Makeham, W. M. (1860). "On the Law of Mortality and the Construction of Annuity Tables". J. Inst. Actuaries and Assur. Mag. 8: 301–310.
^ Gompertz, B. (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society 115: 513–585. doi:10.1098/rstl.1825.0026.
^ ^a ^b ^c Leonid A. Gavrilov & Natalia S. Gavrilova (1991) The Biology of Life Span: A Quantitative Approach. New York: Harwood Academic Publisher, ISBN 3-7186-4983-7
^ Gavrilov, Leonid A.; Gavrilova, Natalia S. (2011). "Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File" (PDF). North American Actuarial Journal: 432–447.
^ Gavrilov, L.A., Gavrilova, N.S., Nosov, V.N. (1983) Human life span stopped increasing: Why? Gerontology, 29(3): 176–180.
^ Gavrilov, L. A.; Nosov, V. N. (1985). "A new trend in human mortality decline: derectangularization of the survival curve". Age 8 (3): 93.
^ Gavrilova N.S., Gavrilov L.A. (2011) Ageing and Longevity: Mortality Laws and Mortality Forecasts for Ageing Populations [In Czech: Stárnutí a dlouhovekost: Zákony a prognózy úmrtnosti pro stárnoucí populace]. Demografie, 53(2): 109–128.
^ Jodrá, P. (2009). "A closed-form expression for the quantile function of the Gompertz–Makeham distribution". Mathematics and Computers in Simulation 79 (10): 3069–3075. doi:10.1016/j.matcom.2009.02.002.

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English Journal

Justifying the Gompertz curve of mortality via the generalized Polya process of shocks.

Cha JH1, Finkelstein M2.
Theoretical population biology.Theor Popul Biol.2016 Mar 14. pii: S0040-5809(16)00014-9. doi: 10.1016/j.tpb.2016.03.001. [Epub ahead of print]
A new probabilistic model of aging that can be applied to organisms is suggested and analyzed. Organisms are subject to shocks that follow the generalized Polya process (GPP), which has been recently introduced and characterized in the literature. Distinct from the nonhomogeneous Poisson process tha
PMID 26988400

Effect of alteplase on the CT hyperdense artery sign and outcome after ischemic stroke.

Mair G, von Kummer R, Morris Z, von Heijne A, Bradey N, Cala L, Peeters A, Farrall AJ, Adami A, Potter G, Cohen G, Sandercock PA, Lindley RI, Wardlaw JM; IST-3 Collaborative Group.
Neurology.Neurology.2016 Jan 12;86(2):118-25. doi: 10.1212/WNL.0000000000002236. Epub 2015 Dec 9.
OBJECTIVE: To investigate whether the location and extent of the CT hyperdense artery sign (HAS) at presentation affects response to IV alteplase in the randomized controlled Third International Stroke Trial (IST-3).METHODS: All prerandomization and follow-up (24-48 hours) CT brain scans in IST-3 we
PMID 26658907

On some mortality rate processes and mortality deceleration with age.

Cha JH1, Finkelstein M2,3.
Journal of mathematical biology.J Math Biol.2016 Jan;72(1-2):331-42. doi: 10.1007/s00285-015-0885-0. Epub 2015 Apr 29.
A specific mortality rate process governed by the non-homogeneous Poisson process of point events is considered and its properties are studied. This process can describe the damage accumulation in organisms experiencing external shocks and define its survival characteristics. It is shown that, altho
PMID 25921379

Japanese Journal

THE MALAYSIAN GOVERNMENT'S ROAD ACCIDENT DEATH REDUCTION TARGET FOR YEAR 2010

LAW T. H.,RADIN UMAR R. S.,WONG S. V.
IATSS research 29(1), 42-49, 2005-05-01
NAID 10019045499

Application of the Logistic, Gompertz, and Richards Growth Functions to Gentan Probability Analysis

Yoshimoto Atsushi
Journal of forest research 6(4), 265-272, 2001-11-16
… In this paper, application of other growth functions, i,e., the logistic, Gompertz and Richards growth functions, is addressed. … This leads to be binomial probability law for a stochastic process, satisfying the unity requirement of the sum of the Gentan probabilities. …
NAID 110002711047

2次元林分遷移の方程式とその応用に関する研究

田中和博
三重大学生物資源学部演習林報告 (17), p1-171, 1992-03
… When we do not consider the randorn fluctuations, the phenomena in the entire growth period of both diameter and height can be described by the differential equation of MITSCHERLICH's growth law. … Fitting the GOMPERTZ growth curve, Equation(180), to the increase of　variance of the diameter distribution in the upper story resulted in a good fit with an upper limit of　about 100 cm2. …
NAID 120000947504

「ゴンペルツの法則」

Gompertz–Makeham
Parameters	(real); (real); (real)
Support
PDF
CDF

　　[★]

英: Gompertz law
同: ゴンパーツの法則、ゴンパーツ仮説ゴンペルツ仮説 Gompertz hypothesis; Gompertz–Makeham law of mortality

[show details]

ゴンパーツ仮説 : 4 件
ゴンペルツ仮説 : 18 件
ゴンパーツの仮説 : 0 件
ゴンペルツの仮説 : 0 件

ヒトの年齢と死亡率との関係を表す法則．35～40歳以降では，年齢に対して死亡率は指数関数的に増大する．35歳以降8年ごとに2倍となるとされている．

参考

Gompertz–Makeham law of mortality - wikipedia en

http://en.wikipedia.org/wiki/Gompertz%E2%80%93Makeham_law_of_mortality

Gompertz function - wikipedia en

http://en.wikipedia.org/wiki/Gompertz_function

原著論文？

On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies

GOMPERTZ B.
Phil. Trans. R. Soc. 115, 513-580, 1825
NAID 30035335012

On the nature of the function expressive of the law of human mortality and on a new mode of determining life contingencies

GOMPERTZ B.
Pholos Trans Royal Soc. (London) 115, 513, 1825
NAID 10008483379

「law」

　　[★]

n.

法、法律

関: jurisprudence、litigation、method

[Makeham1860-1] Makeham, W. M. (1860). "On the Law of Mortality and the Construction of Annuity Tables". J. Inst. Actuaries and Assur. Mag. 8: 301–310.

[Gompertz1825-2] Gompertz, B. (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society 115: 513–585. doi:10.1098/rstl.1825.0026.

[Leonid-3] Leonid A. Gavrilov & Natalia S. Gavrilova (1991) The Biology of Life Span: A Quantitative Approach. New York: Harwood Academic Publisher, ISBN 3-7186-4983-7

[4] Gavrilov, Leonid A.; Gavrilova, Natalia S. (2011). "Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File" (PDF). North American Actuarial Journal: 432–447.

[5] Gavrilov, L.A., Gavrilova, N.S., Nosov, V.N. (1983) Human life span stopped increasing: Why? Gerontology, 29(3): 176–180.

[Gavrilov1985-6] Gavrilov, L. A.; Nosov, V. N. (1985). "A new trend in human mortality decline: derectangularization of the survival curve". Age 8 (3): 93.

[7] Gavrilova N.S., Gavrilov L.A. (2011) Ageing and Longevity: Mortality Laws and Mortality Forecasts for Ageing Populations [In Czech: Stárnutí a dlouhovekost: Zákony a prognózy úmrtnosti pro stárnoucí populace]. Demografie, 53(2): 109–128.

[Jodra2009-8] Jodrá, P. (2009). "A closed-form expression for the quantile function of the Gompertz–Makeham distribution". Mathematics and Computers in Simulation 79 (10): 3069–3075. doi:10.1016/j.matcom.2009.02.002.

v t e Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Gompertz law