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Normal-inverse Gaussian Distribution
The normal-inverse Gaussian distribution (NIG) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997. The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle. Properties Moments The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. Linear transformation This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If : x\sim\mathcal(\alpha, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Location Parameter
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ambiguous boundary, relying more on human or social attributes of place identity and sense of place than on geometry. Types Locality A locality, settlement, or populated place is likely to have a well-defined name but a boundary that is not well defined varies by context. London, for instance, has a legal boundary, but this is unlikely to completely match with general usage. An area within a town, such as Covent Garden in London, also almost always has some ambiguity as to its extent. In geography, location is considered to be more precise than "place". Relative location A relative location, or situation, is described as a displacement from another site. An example is "3 miles northwest of Seattle". Absolute location An absolute lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Divisibility (probability)
In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function.Lukacs, E. (1970) ''Characteristic Functions'', Griffin , London. p. 107 More rigorously, the probability distribution ''F'' is infinitely divisible if, for every positive integer ''n'', there exist ''n'' i.i.d. random variables ''X''''n''1, ..., ''X''''nn'' whose sum ''S''''n'' = ''X''''n''1 + … + ''X''''nn'' has the same distribution ''F''. The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
EM Algorithm
EM, Em or em may refer to: Arts and entertainment Music * EM, the E major musical scale * Em, the E minor musical scale * Electronic music, music that employs electronic musical instruments and electronic music technology in its production * Encyclopedia Metallum, an online metal music database * Eminem, American rapper Other uses in arts and entertainment * ''Em'' (comic strip), a comic strip by Maria Smedstad Companies and organizations * European Movement, an international lobbying association * Aero Benin (IATA code), a defunct airline * Empire Airlines (IATA code), a charter and cargo airline based in Idaho, US * Erasmus Mundus, an international student-exchange program * ExxonMobil, a large oil company formed from the merger of Exxon and Mobil in 1999 * La République En Marche! (sometimes shortened to "En Marche!"), a major French political party Economics * Emerging markets, nations undergoing rapid industrialization Language and typography Language * M, a letter of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ancestral Sampling
An ancestor, also known as a forefather, fore-elder or a forebear, is a parent or ( recursively) the parent of an antecedent (i.e., a grandparent, great-grandparent, great-great-grandparent and so forth). ''Ancestor'' is "any person from whom one is descended. In law, the person from whom an estate has been inherited." Two individuals have a genetic relationship if one is the ancestor of the other or if they share a common ancestor. In evolutionary theory, species which share an evolutionary ancestor are said to be of common descent. However, this concept of ancestry does not apply to some bacteria and other organisms capable of horizontal gene transfer. Some research suggests that the average person has twice as many female ancestors as male ancestors. This might have been due to the past prevalence of polygynous relations and female hypergamy. Assuming that all of an individual's ancestors are otherwise unrelated to each other, that individual has 2''n'' ancestors in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy Processes
Levy, Lévy or Levies may refer to: People * Levy (surname), people with the surname Levy or Lévy * Levy Adcock (born 1988), American football player * Levy Barent Cohen (1747–1808), Dutch-born British financier and community worker * Levy Fidelix (1951–2021), Brazilian conservative politician, businessman and journalist * Levy Gerzberg (born 1945), Israeli-American entrepreneur, inventor, and business person * Levy Li (born 1987), Miss Malaysia Universe 2008–2009 * Levy Mashiane (born 1996), South African footballer * Levy Matebo Omari (born 1989), Kenyan long-distance runner * Levy Mayer (1858–1922), American lawyer * Levy Middlebrooks (born 1966), American basketball player * Levy Mokgothu, South African footballer * Levy Mwanawasa (1948–2008), President of Zambia from 2002 * Levy Nzoungou (born 1998), Congolese-French rugby player, playing in England * Levy Rozman (born 1995), American chess IM, coach, and content creator * Levy Sekgapane (born 1990), South Afri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. It is one of the best known Lévy processes ( càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics. The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Independence
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution Of Probability Distributions
The convolution/sum of probability distributions arises in probability theory and statistics as the operation in terms of probability distributions that corresponds to the addition of independent random variables and, by extension, to forming linear combinations of random variables. The operation here is a special case of convolution in the context of probability distributions. Introduction The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions: see List of convolutions of probability distributions The general formula for the distribution of the sum Z=X+Y of two independent integer-valued ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hyperbolic Distribution
The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution (GIG). Its probability density function (see the box) is given in terms of modified Bessel function of the second kind, denoted by K_\lambda.Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013 It was introduced by Ole Barndorff-Nielsen, who studied it in the context of physics of wind-blown sand. Properties Linear transformation This class is closed under affine transformations. Summation Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible and since the GH distribution can be obtained as a normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution, Barndorff-Nielsen and Halgreen showed the GH distribution is infinite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limit (mathematics), limits, continuous function, continuity and derivatives. The set of real numbers is mathematical notation, denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers subset, include t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
