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Gram–Charlier Theory
In probability theory, the Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution over the real line (-\infty,\infty) in terms of its cumulants. The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ. The key idea of these expansions is to write the characteristic function of the distribution whose probability density function is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover through the inverse Fourier transform. Gram–Charlier A series We examine a continuous random variable. Let \hat be the characteristic function of its distribution whose density function is , and \kappa_r its cumulants. We expand in terms of a known distribution with probability density function , characteristic functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Limit Theorem
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distribution, standard normal distribution. This holds even if the original variables themselves are not Normal distribution, normally distributed. There are several versions of the CLT, each applying in the context of different conditions. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated as late as 1920. In statistics, the CLT can be stated as: let X_1, X_2, \dots, X_n denote a Sampling ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ole Barndorff-Nielsen
Ole Eiler Barndorff-Nielsen (18 March, 1935 – 26 June, 2022) was a Danish statistician who has contributed to many areas of statistical science. Education and career He was born in Copenhagen, and became interested in statistics when, as a student of actuarial mathematics at the University of Copenhagen, he worked part-time at the Department of Biostatistics of the Danish State Serum Institute. He graduated from the University of Aarhus (Denmark) in 1960, where he has spent most of his academic life, and where he became professor of statistics in 1973. However, in 1962-1963 and 1963-1964 he stayed at the University of Minnesota and Stanford University, respectively, and from August 1974 to February 1975 he was an Overseas Fellow at Churchill College, Cambridge, and visitor at Statistical Laboratory, Cambridge University. Barndorff-Nielsen became Professor Emeritus at Aarhus University at the Thiele Centre for Applied Mathematics in Natural Science and affiliated with the '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Cox (statistician)
Sir David Roxbee Cox (15 July 1924 – 18 January 2022) was a British statistician and educator. His wide-ranging contributions to the field of statistics included introducing logistic regression, the proportional hazards model and the Cox process, a point process named after him. He was a professor of statistics at Birkbeck, University of London, Birkbeck College, London, Imperial College London and the University of Oxford, and served as Warden (college), Warden of Nuffield College, Oxford. The first recipient of the International Prize in Statistics, he also received the Guy Medal, Guy, George Box Medal, George Box and Copley Medal, Copley medals, as well as a Knight bachelor, knighthood. Early life Cox was born in Birmingham on 15 July 1924. His father was a die (manufacturing), die sinker and part-owner of a jewellery shop, and they lived near the Jewellery Quarter. The aeronautical engineer Harold Roxbee Cox was a distant cousin. He attended Handsworth Grammar School, B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter McCullagh
Peter McCullagh (born 8 January 1952) is a Northern Irish-born American statistician and John D. MacArthur Distinguished Service Professor in the Department of Statistics at the University of Chicago. Education McCullagh is from Plumbridge, Northern Ireland. He attended the University of Birmingham and completed his PhD at Imperial College London, supervised by David Cox and Anthony Atkinson. Research McCullagh is the coauthor with John Nelder of ''Generalized Linear Models'' (1983, Chapman and Hall – second edition 1989), a seminal text on the subject of generalized linear models (GLMs) with more than 23,000 citations. He also wrote "Tensor Methods in Statistics", published originally in 1987. Awards and honours McCullagh is a Fellow of the Royal Society and the American Academy of Arts and Sciences. He won the COPSS Presidents' Award in 1990. He was the recipient of the Royal Statistical Society's Guy Medal in Bronze in 1983 and in Silver in 2005. He was also th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statistical theory".Kingman 1986, p. 186. Biography Early life Harald Cramér was born in Stockholm, Sweden on 25 September 1893. Cramér remained close to Stockholm for most of his life. He entered the Stockholm University as an undergraduate in 1912, where he studied mathematics and chemistry. During this period, he was a research assistant under the famous chemist, Hans von Euler-Chelpin, with whom he published his first five articles from 1913 to 1914. Following his lab experience, he began to focus solely on mathematics. He eventually began his work on his doctoral studies in mathematics which were supervised by Marcel Riesz at the Stockholm University. Also influenced by G. H. Hardy, Cramér's research led to a PhD in 1917 for his thesi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgeworth Binomial Tree
In quantitative finance, a lattice model is a numerical approach to the valuation of derivatives in situations requiring a discrete time model. For dividend paying equity options, a typical application would correspond to the pricing of an American-style option, where a decision to exercise is allowed at the closing of any calendar day up to the maturity. A continuous model, on the other hand, such as the standard Black–Scholes one, would only allow for the valuation of European options, where exercise is limited to the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, because of path dependence in the payoff. Traditional Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, but some methods now exist for solvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Error
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation derived for it. This inherent error in approximation can be quantified and expressed in two principal ways: as an absolute error, which denotes the direct numerical magnitude of this discrepancy irrespective of the true value's scale, or as a relative error, which provides a scaled measure of the error by considering the absolute error in proportion to the exact data value, thus offering a context-dependent assessment of the error's significance. An approximation error can manifest due to a multitude of diverse reasons. Prominent among these are limitations related to computing machine precision, where digital systems cannot represent all real numbers with perfect accuracy, leading to unavoidable truncation or rounding. Another common source is inherent measurement error, stemming from the practical limitations of inst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Distribution
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: # With a shape parameter and a scale parameter # With a shape parameter \alpha and a rate parameter In each of these forms, both parameters are positive real numbers. The distribution has important applications in various fields, including econometrics, Bayesian statistics, and life testing. In econometrics, the (''α'', ''θ'') parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an Erlang distribution for integer ''α'' values. Bayesian statisticians prefer the (''α'',''λ'') parameterization, utilizing the gamma distribution as a conjugate prior for several inverse scale parameters, facilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgeworth Expansion Of The Density Of The Sample Mean Of Three Chi2 Variables
Edgeworth may refer to: People * Edgeworth (surname) Places * Edgeworth, Gloucestershire, England * Edgeworth, New South Wales, Australia * Edgeworth, Pennsylvania, USA * Edgworth, a village in Lancashire, England * Edgeworth Island, Nunavut, Canada * Edgeworthstown, County Longford, Ireland Other uses * Edgeworth conjecture on the relation of the core and the Walrasian equilibria * Edgeworth series In probability theory, the Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution over th ... of higher-order asymptotic expansions for probability densities. See also * * {{disambiguation, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Statistics
The ''Annals of Statistics'' is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics. It was started in 1973 as a continuation in part of the '' Annals of Mathematical Statistics (1930)'', which was split into the ''Annals of Statistics'' and the '' Annals of Probability''. The journal CiteScore is 5.8, and its SCImago Journal Rank is 5.877, both from 2020. Articles older than 3 years are available on JSTOR, and all articles since 2004 are freely available on the arXiv arXiv (pronounced as "archive"—the X represents the Chi (letter), Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly pee .... Editorial board The following persons have been editors of the journal: * Ingram Olkin (1972–1973) * I. Richard Savage (1974–1976) * Rupert G. Miller (1977–1979) * David V. Hinkley (1980–1982) * Michael D. Perlm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
