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Nelson–Aalen Estimator
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 the failure of a non-repairable component, the death of a human being, or any occurrence for which the experimental unit remains in the "failed" state (e.g., death) from the point at which it changed on. The estimator is given by :\tilde(t)=\sum_\frac, with d_i the number of events at t_i and n_i the total individuals at risk at t_i. 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 homogeneity of Poisson processes. It was constructed by Wayne Nelson and Odd Aalen. See also * Kaplan–Meier estimator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonparametric Statistics
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distribution-free or having a specified distribution but with the distribution's parameters unspecified. Nonparametric statistics includes both descriptive statistics and statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are violated. Definitions The term "nonparametric statistics" has been imprecisely defined in the following two ways, among others: Applications and purpose Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 analysis in engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ..., duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival? To answer such q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Censoring (statistics)
In statistics, censoring is a condition in which 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 rate. In such a study, it may be known that an individual's age at death is ''at least'' 75 years (but may be more). Such a situation could occur if the individual withdrew from the study at age 75, or if the individual is currently alive at the age of 75. Censoring also occurs when a value occurs outside the range of a measuring instrument. For example, a bathroom scale might only measure up to 140 kg. If a 160-kg individual is weighed using the scale, the observer would only know that the individual's weight is at least 140 kg. The problem of censored data, in which the observed value of some variable is partially known, is related to the problem of missing data, where the observed value of some variable is unknown. Censoring should not be confused with the related idea tru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Missing Data
In statistics, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data. Missing data can occur because of nonresponse: no information is provided for one or more items or for a whole unit ("subject"). Some items are more likely to generate a nonresponse than others: for example items about private subjects such as income. Attrition is a type of missingness that can occur in longitudinal studies—for instance studying development where a measurement is repeated after a certain period of time. Missingness occurs when participants drop out before the test ends and one or more measurements are missing. Data often are missing in research in economics, sociology, and political science because governments or private entities choose not to, or fail to, report critical statistics, or because the information is not availab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Survival Analysis
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 analysis in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival? To answer such questions, it is necessary to define "lifetime". In the case of biological survival, death is unambiguous, but for mechanical reliability, failure may not be well-defined, for there may well be mechanical systems in which failure is partial, a matter of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reliability Engineering
Reliability engineering is a sub-discipline of systems engineering that emphasizes the ability of equipment to function without failure. Reliability describes the ability of a system or component to function under stated conditions for a specified period of time. Reliability is closely related to availability, which is typically described as the ability of a component or system to function at a specified moment or interval of time. The reliability function is theoretically defined as the probability of success at time t, which is denoted R(t). This probability is estimated from detailed (physics of failure) analysis, previous data sets or through reliability testing and reliability modelling. Availability, testability, maintainability and maintenance, repair and operations, maintenance are often defined as a part of "reliability engineering" in reliability programs. Reliability often plays the key role in the cost-effectiveness of systems. Reliability engineering deals with the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Life Insurance
Life insurance (or life assurance, especially in the Commonwealth of Nations) is a contract between an insurance policy holder and an insurer or assurer, where the insurer promises to pay a designated beneficiary a sum of money upon the death of an insured person (often the policyholder). Depending on the contract, other events such as terminal illness or critical illness can also trigger payment. The policyholder typically pays a premium, either regularly or as one lump sum. The benefits may include other expenses, such as funeral expenses. Life policies are legal contracts and the terms of each contract describe the limitations of the insured events. Often, specific exclusions written into the contract limit the liability of the insurer; common examples include claims relating to suicide, fraud, war, riot, and civil commotion. Difficulties may arise where an event is not clearly defined, for example, the insured knowingly incurred a risk by consenting to an experimental m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the sample mean is a commonly used estimator of the population mean. There are point and interval estimators. The point estimators yield single-valued results. This is in contrast to an interval estimator, where the result would be a range of plausible values. "Single value" does not necessarily mean "single number", but includes vector valued or function valued estimators. ''Estimation theory'' is concerned with the properties of estimators; that is, with defining properties that can be used to compare different estimators (different rules for creating estimates) for the same quantity, based on the same data. Such properties can be used to determine the best rules to use under given circumstances. However, in robust statistics, statistica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bathtub Curve
The bathtub curve is widely used in reliability engineering and deterioration modeling. 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. *The second part is a constant failure rate, known as random failures. *The third part is an increasing failure rate, known as wear-out failures. The name is derived from the cross-sectional shape of a bathtub: steep sides and a flat bottom. The bathtub curve is generated by mapping the rate of early "infant mortality" failures when first introduced, the rate of random failures with constant failure rate during its "useful life", and finally the rate of "wear out" failures as the product exceeds its design lifetime. In less technical terms, in the early life of a product adhering to the bathtub curve, the failure rate is high but rapidly decreasing as defective products are identified and discarded, and early sources of potential failure s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996. biology,H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988. ecology,H. Thompson. Spatial point processes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wayne Nelson (statistician)
''For the American musician, see Wayne Nelson.'' Wayne Nelson is an American statistician. His main contributions to the reliability theory are the Nelson-Aalen Estimator for lifetime data, various statistical procedures for accelerated life testing and both: nonparametric and parametric procedures for recurrent data analysis. Early life and education Nelson was born in Chicago in 1936. He studied Physics at Caltech and graduated with a Bachelor of Science in 1958. Nelson obtained a Master of Science in Physics from the University of Illinois in 1959, then a Ph.D. in statistics from the same university in 1965. Career Nelson was employed from 1965 to 1989 at General Electric R&D. He was also an adjunct professor teaching graduate courses on applications of statistics at Union College and Rensselaer Polytechnic Institute. Currently, Nelson works as a private consultant and legal expert witness in statistical analysis and modeling of data in many industries; including automo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Aalen
Odd Olai Aalen (born 6 May 1947, in Oslo) is a Norwegian statistician and a professor at the Department of Biostatistics at the Institute of Basic Medical Sciences at the University of Oslo. Life Aalen completed his examen artium in 1966 at Oslo Cathedral School before studying first mathematics and physics and then statistics in which he graduated at the University of Oslo in 1972. Work His research work is geared towards applications in biosciences. Aalen's early work on counting processes and martingales, starting with his 1976 Ph.D. thesis at the University of California, Berkeley, has had profound influence in biostatistics. Inferences for fundamental quantities associated with cumulative hazard rates, in survival analysis and models for analysis of event histories, are typically based on the Nelson–Aalen estimator or appropriate related statistics. The Nelson–Aalen estimator is related to the Kaplan-Meier estimator and generalisations thereof. Aalen is currently prof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
