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Asymptotic Distribution
In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the limiting distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical estimators. Definition A sequence of distributions corresponds to a sequence of random variables ''Zi'' for ''i'' = 1, 2, ..., I . In the simplest case, an asymptotic distribution exists if the probability distribution of ''Zi'' converges to a probability distribution (the asymptotic distribution) as ''i'' increases: see convergence in distribution. A special case of an asymptotic distribution is when the sequence of random variables is always zero or ''Zi'' = 0 as ''i'' approaches infinity. Here the asymptotic distribution is a degenerate distribution, corresponding to the value zero. However, the most usual sense in which the term asymptotic distribution is used arises ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence In Distribution
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution. The concept is important in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delta Method
In statistics, the delta method is a method of deriving the asymptotic distribution of a random variable. It is applicable when the random variable being considered can be defined as a differentiable function of a random variable which is Asymptotic distribution, asymptotically Normal distribution, Gaussian. History The delta method was derived from propagation of error, and the idea behind was known in the early 20th century. Its statistical application can be traced as far back as 1928 by Truman Lee Kelley, T. L. Kelley. A formal description of the method was presented by J. L. Doob in 1935. Robert Dorfman also described a version of it in 1938. Univariate delta method While the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, if there is a sequence (mathematics), sequence of random variables satisfying :, where ''θ'' and ''σ''2 are finite valued constants and \xrightar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limiting Density Of Discrete Points
In information theory, the limiting density of discrete points is an adjustment to the formula of Claude Shannon for differential entropy. It was formulated by Edwin Thompson Jaynes to address defects in the initial definition of differential entropy. Definition Shannon originally wrote down the following formula for the entropy of a continuous distribution, known as differential entropy: : h(X)=-\int p(x)\log p(x)\,dx. Unlike Shannon's formula for the discrete entropy, however, this is not the result of any derivation (Shannon simply replaced the summation symbol in the discrete version with an integral), and it lacks many of the properties that make the discrete entropy a useful measure of uncertainty. In particular, it is not invariant under a change of variables and can become negative. In addition, it is not even dimensionally correct. Since h(X) would be dimensionless, p(x) must have units of \frac , which means that the argument to the logarithm is not dimensionless ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Moivre–Laplace Theorem
In probability theory, the de Moivre–Laplace theorem, which is a special case of the central limit theorem, states that the normal distribution may be used as an approximation to the binomial distribution under certain conditions. In particular, the theorem shows that the probability mass function of the random number of "successes" observed in a series of n independent Bernoulli trials, each having probability p of success (a binomial distribution with n trials), converges to the probability density function of the normal distribution with expectation np and standard deviation \sqrt, as n grows large, assuming p is not 0 or 1. The theorem appeared in the second edition of '' The Doctrine of Chances'' by Abraham de Moivre, published in 1738. Although de Moivre did not use the term "Bernoulli trials", he wrote about the probability distribution of the number of times "heads" appears when a coin is tossed 3600 times. This is one derivation of the particular Gaussian function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Theory (statistics)
In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size may grow indefinitely; the properties of estimators and tests are then evaluated under the limit of . In practice, a limit evaluation is considered to be approximately valid for large finite sample sizes too.Höpfner, R. (2014), Asymptotic Statistics, Walter de Gruyter. 286 pag. , Overview Most statistical problems begin with a dataset of size . The asymptotic theory proceeds by assuming that it is possible (in principle) to keep collecting additional data, thus that the sample size grows infinitely, i.e. . Under the assumption, many results can be obtained that are unavailable for samples of finite size. An example is the weak law of large numbers. The law states that for a sequence of independent and identically distributed (IID) random variables , if one value is drawn from each ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing Limit (mathematics), limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to . Then the theorem states that \pi(x)\sim\frac. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. Definition Formally, given functions and , we define a binary relation f( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chapman And Hall
Chapman & Hall is an imprint owned by CRC Press, originally founded as a British publishing house in London in the first half of the 19th century by Edward Chapman and William Hall. Chapman & Hall were publishers for Charles Dickens (from 1840 until 1844 and again from 1858 until 1870), Thomas Carlyle, William Thackeray, Elizabeth Barrett Browning, Anthony Trollope, Eadweard Muybridge and Evelyn Waugh. History Upon Hall's death in 1847, Chapman's cousin Frederic Chapman began his progress through the ranks of the company and eventually becoming a partner in 1858 and sole proprietor on Edward Chapman's retirement from Chapman & Hall in 1866. In 1868 author Anthony Trollope bought a third of the company for his son, Henry Merivale Trollope. From 1902 to 1930 the company's managing director was Arthur Waugh. In the 1930s the company merged with Methuen, a merger which, in 1955, participated in forming the Associated Book Publishers. The latter was acquired by The Thomson ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Parametric Model
Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, a main character who appears more frequently and/or prominently than a recurring character * Regular division of the plane, a series of drawings by the Dutch artist M. C. Escher which began in 1936 Language * Regular inflection, the formation of derived forms such as plurals in ways that are typical for the language ** Regular verb * Regular script, the newest of the Chinese script styles Mathematics Algebra and number theory * Regular category, a kind of category that has similarities to both Abelian categories and to the category of sets * Regular chains in computer algebra * Regular element (other), certain kinds of elements of an algebraic structure * Regular extension of fields * Regular ideal (multiple definitions) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model represents, often in considerably idealized form, the Data generating process, data-generating process. When referring specifically to probability, probabilities, the corresponding term is probabilistic model. All Statistical hypothesis testing, statistical hypothesis tests and all Estimator, statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman J. Adèr, Herman Adèr quoting Kenneth A. Bollen, Kenneth Bollen). Introduction Informally, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, Academic journal, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, Technology, technical, and engineering subject areas, abandoning its literary interests. Wiley's son Joh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed
Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist group Music Groups, labels, and genres * Independent music, a number of genres associated with independent labels * Independent record label, a record label not associated with a major label * Independent Albums, American albums chart Albums * ''Independent'' (Ai album), 2012 * ''Independent'' (Faze album), 2006 * ''Independent'' (Sacred Reich album), 1993 Songs * "Independent" (song), a 2007 song by Webbie * "Independent", a 2002 song by Ayumi Hamasaki from '' H'' News media organizations * Independent Media Center (also known as Indymedia or IMC), an open publishing network of journalist collectives that report on political and social issues, e.g., in ''The Indypendent'' newspaper of NYC * ITV (TV network) (Independent Television ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
