en>BattyBot: Converted {{Multiple issues}} to new format to fix expert parameter & general fixes using AWB (8853)

2013-03-06T16:41:33Z

Converted {{Multiple issues}} to new format to fix expert parameter & general fixes using AWB (8853)

← Older revision		Revision as of 18:41, 6 March 2013
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	~~Andera~~ is ~~what you can call~~ [http://~~test~~.~~jeka~~-nn.ru/~~node~~/~~129 real psychics~~] ~~her but she by no means really~~ ~~email psychic readings~~ ([http://~~165.132~~.39.93/xe/~~visitors~~/~~372912 http:~~//~~165~~.~~132~~.39.93/~~xe/visitors/372912]~~) ~~liked that title~~. ~~For years he's been living~~ in ~~Alaska~~ and ~~he doesn't strategy on changing it~~. ~~The favorite hobby for him~~ and ~~his kids is style~~ and ~~he'll be starting some thing else along with it~~. He is ~~an information officer.~~<br><br>~~My web~~-~~site; clairvoyance (~~[~~http~~:/~~/hknews~~.~~classicmall~~.~~com~~.hk/~~groups/some~~-~~simple~~-~~tips~~-~~for~~-~~personal~~-~~development-progress~~/ ~~http:~~/~~/hknews~~.~~classicmall.com.hk~~/~~groups/some~~-~~simple~~-~~tips~~-~~for-personal-development-progress~~])		{{Probability distribution
			\| name = Gompertz
			Makeham
			\| type = continuous
			\| pdf_image =
			\| cdf_image =
			\| notation =
			\| parameters = <math>\alpha > 0</math> ([[real number\|real]])<br/><math>\beta > 0</math> (real)<br/> <math>\lambda > 0</math> (real)
			\| support = <math>x \in \mathbb{R}^+</math>
			\| pdf = <math>(\alpha e^{\beta x} + \lambda)\cdot \exp(-\lambda x-\frac{\alpha}{\beta}(e^{\beta x}-1))</math>
			\| cdf = <math>1-\exp(-\lambda x-\frac{\alpha}{\beta}(e^{\beta x}-1))</math>
			\| mean =
			\| median =
			\| mode =
			\| variance =
			\| skewness =
			\| kurtosis =
			\| entropy =
			\| mgf =
			\| cf =
			\| pgf =
			\| fisher =
			}}

			The '''Gompertz–Makeham law'''
			states that the death rate is the sum of an age-independent component (the Makeham term, named after [[William Makeham]])<ref name="Makeham1860">{{cite journal \|last=Makeham \|first=W. M. \|authorlink= \|coauthors= \|year=1860 \|month= \|title=On the Law of Mortality and the Construction of Annuity Tables \|journal=J. Inst. Actuaries and Assur. Mag. \|volume=8 \|issue= \|pages=301–310}}</ref> and an age-dependent component (the [[Gompertz function]], named after [[Benjamin Gompertz]]),<ref name="Gompertz1825">{{cite journal \|last=Gompertz \|first=B. \|authorlink= \|coauthors= \|year=1825 \|month= \|title=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 \|journal=Philosophical Transactions of the Royal Society \|volume=115 \|issue= \|pages=513–585 \|id= \|url=http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-55920 Â\|doi=10.1098/rstl.1825.0026}}
			</ref> which [[exponential growth\|increases exponentially]] with age.<ref name="Leonid">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</ref> 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]]<ref name="Leonid" /> – but other studies disagree.

			[[Image:USGompertzCurve.svg\|thumb\|Estimated probability of a person dying at each age, for the U.S. in 2003 [http://www.cdc.gov/nchs/data/nvsr/nvsr54/nvsr54_14.pdf]. Mortality rates increase exponentially with age after age 20.]]

			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.<ref name="Leonid" /><ref>Gavrilov, L.A., Gavrilova, N.S., Nosov, V.N. (1983) Human life span stopped increasing: Why? Gerontology, 29(3): 176–180.</ref> Since the 1950s, a new mortality trend has started in the form of an unexpected decline in mortality rates at advanced ages and "de-rectangularization" of the survival curve.<ref name="Gavrilov1985">{{cite journal \|last=Gavrilov \|first=L. A. \|authorlink= \|coauthors=Nosov, V. N. \|year=1985 \|month= \|title=A new trend in human mortality decline: derectangularization of the survival curve \|journal=Age \|volume=8 \|issue=3 \|pages=93}}</ref><ref>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.</ref>

			The hazard function for the Gompert-Makeham distribution is most often characterised as <math>h(x)=\alpha e^{\beta x} + \lambda </math>

			The [[quantile function]] can be expressed in a closed-form expressions using the [[Lambert W function]]:<ref name="Jodra2009">{{cite journal \|last=Jodrá \|first=P. \|authorlink= \|coauthors= \|year=2009 \|month= \|title=A closed-form expression for the quantile function of the Gompertz–Makeham distribution \|journal=Mathematics and Computers in Simulation \|volume=79 \|issue= 10\|pages=3069–3075 \|id= \|url= \|doi=10.1016/j.matcom.2009.02.002}}</ref>

			: <math>Q(u)=\frac{\alpha}{\beta\lambda}-\frac{1}{\lambda} \ln(1-u)-\frac{1}{\beta}W_0\left(\frac{\alpha e^{\alpha/\lambda}(1-u)^{-(\beta/\lambda)}}{\lambda}\right)</math>

			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).

			==See also==
			* [[Biodemography]]
			* [[Biodemography of human longevity]]
			* [[Gerontology]]
			* [[Demography]]
			* [[Life table]]
			* [[Maximum life span]]
			* [[Reliability theory of aging and longevity]]

			==References==
			<references/>
			{{ProbDistributions\|continuous-semi-infinite}}
			{{DEFAULTSORT:Gompertz-Makeham Law Of Mortality}}
			[[Category:Actuarial science]]
			[[Category:Ageing]]
			[[Category:Death]]
			[[Category:Demography]]
			[[Category:Population]]
			[[Category:Senescence]]
			[[Category:Statistical laws]]
			[[Category:Continuous distributions]]
			[[Category:Probability distributions]]

en>Brad7777: removed Category:Dynamical systems using HotCat

2012-02-05T15:26:08Z

removed Category:Dynamical systems using HotCat

New page

Andera is what you can call [http://test.jeka-nn.ru/node/129 real psychics] her but she by no means really email psychic readings ([http://165.132.39.93/xe/visitors/372912 http://165.132.39.93/xe/visitors/372912]) liked that title. For years he's been living in Alaska and he doesn't strategy on changing it. The favorite hobby for him and his kids is style and he'll be starting some thing else along with it. He is an information officer.<br><br>My web-site; clairvoyance ([http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress])

Chetayev instability theorem - Revision history

en>BattyBot: Converted {{Multiple issues}} to new format to fix expert parameter & general fixes using AWB (8853)

en>Brad7777: removed Category:Dynamical systems using HotCat