|
Statistical Functional
V-statistics are a class of statistics named for Richard von Mises who developed their asymptotic distribution theory in a fundamental paper in 1947. V-statistics are closely related to U-statistics (U for " unbiased") introduced by Wassily Hoeffding in 1948. A V-statistic is a statistical function (of a sample) defined by a particular statistical functional of a probability distribution. Statistical functions Statistics that can be represented as functionals T(F_n) of the empirical distribution function (F_n) are called ''statistical functionals''. Differentiability of the functional ''T'' plays a key role in the von Mises approach; thus von Mises considers ''differentiable statistical functionals''. Examples of statistical functions The ''k''-th central moment is the ''functional'' T(F)=\int(x-\mu)^k \, dF(x), where \mu = E /math> is the expected value of ''X''. The associated ''statistical function'' is the sample ''k''-th central moment, : T_n=m_k=T(F_n) = \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Von Mises
Richard Edler von Mises (; 19 April 1883 – 14 July 1953) was an Austrian scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory. He held the position of Gordon McKay Professor of Aerodynamics and Applied Mathematics at Harvard University. He described his work in his own words shortly before his death as being on :"... practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability calculus, statistics and philosophy." Although best known for his mathematical work, von Mises also contributed to the philosophy of science as a neo-positivist and empiricist, following the line of Ernst Mach. Historians of the Vienna Circle of logical empiricism recognize a "first phase" from 1907 through 1914 with Philipp Frank, Hans Hahn, and Otto Neurath. His older brother, Ludwig von Mises, held an opposite point of view with respect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Normal
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 distrib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Normality
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 arise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. 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 general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory. If X_1, X_2, \dots, X_n, \dots are random samples drawn from a population with overall mean \mu and finite variance and if \bar_n is the sample mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chi-squared Distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the central chi-squared distribution, a special case of the more general noncentral chi-squared distribution. The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tes ... [...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] |
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 ar ... [...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] |
