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Edgeworth Expansion
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 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 function \hat, and cumulants \gamma_r. The density is generally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jørgen Pedersen Gram
Jørgen Pedersen Gram (27 June 1850 – 29 April 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark. Important papers of his include ''On series expansions determined by the methods of least squares'', and ''Investigations of the number of primes less than a given number''. The mathematical method that bears his name, the Gram–Schmidt process, was first published in the former paper, in 1883. For number theorists his main fame is the series for the Riemann zeta function (the leading function in Riemann's exact prime-counting function). Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers. It has recently been supplanted by a formula of Ramanujan that uses the Bernoulli numbers directly instead of the zeta function. In control theory, the Gramian or Gram matrix is an important contribution named after him. The C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in the ... [...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 th ... [...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 College, London, Imperial College London and the University of Oxford, and served as Warden of Nuffield College, Oxford. The first recipient of the International Prize in Statistics, he also received the Guy, George Box and Copley medals, as well as a knighthood. Early life Cox was born in Birmingham on 15 July 1924. His father was a 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, Birmingham. He received a Master of Arts in mathematics at St John's College, Cambridge, and obtained his PhD from the Univers ... [...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 als ... [...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 University of Stockholm 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 University of Stockholm. Also influenced by G. H. Hardy, Cramér's research led to a PhD in 1917 for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgeworth Binomial Tree
In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on 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, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem. Equity and commodity derivatives In general the approach is to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cornish–Fisher Expansion
The Cornish–Fisher expansion is an asymptotic expansion used to approximate the quantiles of a probability distribution based on its cumulants. It is named after E. A. Cornish and R. A. Fisher, who first described the technique in 1937. Definition For a random variable ''X'' with mean μ, variance σ², and cumulants κ''n'', its quantile ''yp'' at order-of-quantile ''p'' can be estimated as y_p \approx \mu + \sigma w_p where: : \begin w_p &=& x &+ \left gamma_1 h_1(x)\right\ &&&+ \left gamma_2 h_2(x) + \gamma_1^2 h_(x)\right\ &&&+ \left gamma_3 h_3(x) + \gamma_1\gamma_2 h_(x) + \gamma_1^3 h_(x)\right\ &&&+ \cdots\\ \end : \begin x &= \Phi^(p)\\ \gamma_ &= \frac;\; r \in \\\ h_1(x) &= \frac\\ h_2(x) &= \frac\\ h_(x) &= -\frac\\ h_3(x) &= \frac\\ h_(x) &= -\frac\\ h_(x) &= \frac \end where He''n'' is the ''n''th probabilists' Hermite polynomial. The values ''γ''1 and ''γ''2 are the random variable's skewness and (excess) kurtosis In probability theory and statistic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Error
The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value). An approximation error can occur because of computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.1 cm, so you measure it as 4.5 cm). In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates how the error is propagated by the algorithm. Formal definition One commonly distinguishes between the relative error and the absolute error. Given some value ''v'' and its approximation ''v''approx, the absolute error is :\epsilon = , v-v_\text, \ , where the vertical bars denote the absolute value. If v \ne 0, the relative error is : \eta = \frac = \left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: #With a shape parameter k and a scale parameter \theta. #With a shape parameter \alpha = k and an inverse scale parameter \beta = 1/ \theta , called a rate parameter. In each of these forms, both parameters are positive real numbers. The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a 1/x base measure) for a random variable X for which E 'X''= ''kθ'' = ''α''/''β'' is fixed and greater than zero, and E n(''X'')= ''ψ''(''k'') + ln(''θ'') = ''ψ''(''α'') − ln(''β'') is fixed (''ψ'' is the digamma function). Definitions The parameterization with ''k'' and ''θ'' appears to be more common i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgeworth Expansion Of The Density Of The Sample Mean Of Three Chi2 Variables , a village in Lancashire, England
{{disambiguation, geo ...
Edgeworth may refer to: People * Edgeworth (surname) Places * Edgeworth, Gloucestershire, England * Edgeworth, New South Wales, Australia * Edgeworth, Pennsylvania, USA * Edgeworth Island, Nunavut, Canada * Edgeworthstown, County Longford, Republic of Ireland Other uses * Edgeworth conjecture on the relation of the core and the Walrasian equilibria * Edgeworth series of higher-order asymptotic expansions for probability densities. See also * * * Edgworth Edgworth is a small village within the borough of Blackburn with Darwen, Lancashire, England. It is north east of North Turton between Broadhead Brook on the west (expanded artificially to form the Wayoh Reservoir) and Quarlton Brook in the sou ... [...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. Editorial board The following persons have been editors of the journal: * Ingram Olkin (1972–1973) * I. Richard Savage (1974–1976) * Rupert Miller (1977–1979) * David V. Hinkley (1980–1982) * Michael D. Perlman (1983–1985) * Willem van Zwet (1986–1988) * Arthur Cohen (1988–1991) * Michael Woodroofe (1992–1994) * Larry Brown and John Rice (1995–1997) * Hans-Rudolf Künsch and James O. Berger (1998–2000) * John Marden and Jon A. Wellner (2001–2003) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
