Nelson–Aalen estimator
Encyclopedia
The Nelson–Aalen estimator is a non-parametric estimator
of the cumulative hazard rate function in case of censored data
or incomplete data. It is used in survival theory
, reliability engineering
and life insurance
to estimate the cumulative number of expected events. An event can be a failure of a non-repairable component or a death of a human being. The estimator
is given by
with
the number of events at 
and 
are the total individuals at risk at 
.
The curvature of the Nelson–Aalen estimator gives an idea of the hazard rate shape. A concave shape is an indicator for infant mortality
while a convex shape indicates wear out mortality
.
It can be used for example when testing the homogenitity of Poisson process
es.
Non-parametric statistics
In statistics, the term non-parametric statistics has at least two different meanings:The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:...
of the cumulative hazard rate function in case of censored data
Censoring (statistics)
In statistics, engineering, and medical research, censoring occurs when the value of a measurement or observation is only partially known.For example, suppose a study is conducted to measure the impact of a drug on mortality. In such a study, it may be known that an individual's age at death is at...
or incomplete data. It is used in survival theory
Survival analysis
Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, and duration analysis or duration modeling in economics or sociology...
, reliability engineering
Reliability engineering
Reliability engineering is an engineering field, that deals with the study, evaluation, and life-cycle management of reliability: the ability of a system or component to perform its required functions under stated conditions for a specified period of time. It is often measured as a probability of...
and life insurance
Life insurance
Life insurance is a contract between an insurance policy holder and an insurer, where the insurer promises to pay a designated beneficiary a sum of money upon the death of the insured person. Depending on the contract, other events such as terminal illness or critical illness may also trigger...
to estimate the cumulative number of expected events. An event can be a failure of a non-repairable component or a death of a human being. The estimator
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
is given by
with
the number of events at and are the total individuals at risk at .The curvature of the Nelson–Aalen estimator gives an idea of the hazard rate shape. A concave shape is an indicator for infant mortality
Bathtub curve
The bathtub curve is widely used in reliability engineering. It describes a particular form of the hazard function which comprises three parts:*The first part is a decreasing failure rate, known as early failures....
while a convex shape indicates wear out mortality
Bathtub curve
The bathtub curve is widely used in reliability engineering. It describes a particular form of the hazard function which comprises three parts:*The first part is a decreasing failure rate, known as early failures....
.
It can be used for example when testing the homogenitity of Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
es.
External links
- http://www.statsdirect.com/help/survival_analysis/kaplan.htm
- http://tkchen.wordpress.com/2008/09/21/kaplan-meier-and-nelson-aalen-estimators/