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Glivenko–Cantelli Theorem
In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators. Statement Assume that X_1,X_2,\dots are independent and identically distributed random variables in \mathbb with common cumulative distribution function F(x). The ''empirical distribution function'' for X_1,\dots,X_n is defined by :F_n(x)=\frac\sum_^n I_(x) = \frac\left, \left\\ where I_C is the indicator functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Probability
Probability theory 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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Process
In physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals the time average. In this regime, any collection of random samples from a process must represent the average statistical properties of the entire regime. Conversely, a process that is not in ergodic regime is said to be in non-ergodic regime. Specific definitions One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process X(t) has constant mean :\mu_X= E (t)/math>, and autocovariance :r_X(\tau) = E X(t)-\mu_X) (X(t+\tau)-\mu_X)/math>, that depends only on the lag \tau and not on time t. The properties \mu_X and r_X(\tau) are ''ensemble averages'' (calculated over all possible sample functions X), not time averages. The process X(t) is said to be mean-ergodicPapoulis, p.428 or mean-square ergodic in the first momentPorat, p.14 if the time average estima ... [...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 rand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Process
In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state. For a process in a discrete state space a population continuous time Markov chain or Markov population model is a process which counts the number of objects in a given state (without rescaling). In mean field theory, limit theorems (as the number of objects becomes large) are considered and generalise the central limit theorem for empirical measures. Applications of the theory of empirical processes arise in non-parametric statistics. Definition For ''X''1, ''X''2, ... ''X''''n'' independent and identically-distributed random variables in R with common cumulative distribution function ''F''(''x''), the empirical distribution function is defined by :F_n(x)=\frac\sum_^n I_(X_i), where I''C'' is the indicator function of the set ''C''. For every (fixed) ''x'', ''F''''n''(''x'') is a sequence of random variables which converge to ''F''(''x'') almost ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dvoretzky–Kiefer–Wolfowitz Inequality
In the theory of probability and statistics, the Dvoretzky–Kiefer–Wolfowitz–Massart inequality (DKW inequality) bounds how close an empirically determined distribution function will be to the distribution function from which the empirical samples are drawn. It is named after Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz, who in 1956 proved the inequality : \Pr\Bigl(\sup_ , F_n(x) - F(x), > \varepsilon \Bigr) \le Ce^\qquad \text\varepsilon>0. with an unspecified multiplicative constant ''C'' in front of the exponent on the right-hand side. In 1990, Pascal Massart proved the inequality with the sharp constant ''C'' = 2, confirming a conjecture due to Birnbaum and McCarty. In 2021, Michael Naaman proved the multivariate version of the DKW inequality and generalized Massart's tightness result to the multivariate case, which results in a sharp constant of twice the number of variables, ''C'' = 2k. The DKW inequality Given a natural nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donsker's Theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem. Let X_1, X_2, X_3, \ldots be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let S_n:=\sum_^n X_i. The stochastic process S:=(S_n)_ is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by : W^(t) := \frac, \qquad t\in ,1 The central limit theorem asserts that W^(1) converges in distribution to a standard Gaussian random variable W(1) as n\to\infty. Donsker's invariance principle extends this convergence to the whole function W^:=(W^(t))_. More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space \mathcal ,1/math>, the random function W^ converges in distribution to a standard Brownian moti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atom (measure Theory)
In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless. Definition Given a measurable space (X, \Sigma) and a measure \mu on that space, a set A\subset X in \Sigma is called an atom if \mu(A) > 0 and for any measurable subset B \subset A with \mu(B) of A are atoms, and /math> is called an atomic class. If \mu is a \sigma-finite measure, there are countably many atomic classes. Examples * Consider the set ''X'' = and let the sigma-algebra \Sigma be the power set of ''X''. Define the measure \mu of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons , for ''i'' = 1, 2, ..., 9, 10 is an atom. * Consider the Lebesgue measure on the real line. This measure has no atoms. Atomic measures A \sigma-finite measure \mu on a measurable space (X, \Sigma) is called atomic or pu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shattering (machine Learning)
The concept of shattered sets plays an important role in Vapnik–Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes as well as in statistical computational learning theory. Definition Suppose ''A'' is a set and ''C'' is a class of sets. The class ''C'' shatters the set ''A'' if for each subset ''a'' of ''A'', there is some element ''c'' of ''C'' such that : a = c \cap A. Equivalently, ''C'' shatters ''A'' when their intersection is equal to ''As power set: ''P''(''A'') = . We employ the letter ''C'' to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set ''A'' is often assumed to be finite because, in empirical processes, we are interested in the shattering of finite sets of data points. Example We will show that the class of all discs in the plane (two-dimensional space) does not shatter every set of four points on the unit circle, yet the class of al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chervonenkis
Alexey Yakovlevich Chervonenkis (russian: link=no, Алексей Яковлевич Червоненкис; 7 September 1938 – 22 September 2014) was a Soviet and Russian mathematician. Along with Vladimir Vapnik, he was one of the main developers of the Vapnik–Chervonenkis theory, also known as the "fundamental theory of learning" - an important part of computational learning theory. Chervonenkis held joint appointments with the Russian Academy of Sciences and Royal Holloway, University of London. Alexey Chervonenkis got lost in Losiny Ostrov National Park on 22 September 2014, and later during a search operation was found dead near Mytishchi, a suburb of Moscow. He had died of hypothermia Hypothermia is defined as a body core temperature below in humans. Symptoms depend on the temperature. In mild hypothermia, there is shivering and mental confusion. In moderate hypothermia, shivering stops and confusion increases. In severe .... References External linksChervo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vapnik
Vladimir Naumovich Vapnik (russian: Владимир Наумович Вапник; born 6 December 1936) is one of the main developers of the Vapnik–Chervonenkis theory of statistical learning, and the co-inventor of the support-vector machine method, and support-vector clustering algorithm. Early life and education Vladimir Vapnik was born to a Jewish family in the Soviet Union. He received his master's degree in mathematics from the Uzbek State University, Samarkand, Uzbek SSR in 1958 and Ph.D in statistics at the Institute of Control Sciences, Moscow in 1964. He worked at this institute from 1961 to 1990 and became Head of the Computer Science Research Department. Academic career At the end of 1990, Vladimir Vapnik moved to the USA and joined the Adaptive Systems Research Department at AT&T Bell Labs in Holmdel, New Jersey. While at AT&T, Vapnik and his colleagues did work on the support-vector machine, which he also worked on much earlier before moving to the USA. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the unit interval , 1 returning 0 for the empty set and 1 for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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