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Multivariate Student Distribution
In statistics, the multivariate ''t''-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's ''t''-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix ''t''-distribution is distinct and makes particular use of the matrix structure. Definition One common method of construction of a multivariate ''t''-distribution, for the case of p dimensions, is based on the observation that if \mathbf y and u are independent and distributed as N(,) and \chi^2_\nu (i.e. multivariate normal and chi-squared distributions) respectively, the matrix \mathbf\, is a ''p'' × ''p'' matrix, and /\sqrt = -, then has the density : \frac\left +\frac(-)^T^(-)\right and is said to be distributed as a multivariate ''t''-distribution with parameters ,,\nu. Note that \mathbf\Sigma is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Location Parameter
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ambiguous boundary, relying more on human or social attributes of place identity and sense of place than on geometry. Types Locality A suburb, locality, human settlement, settlement, or populated place is likely to have a well-defined name but a boundary that is not well defined varies by context. London, for instance, has a legal boundary, but this is unlikely to completely match with general usage. An area within a town, such as Covent Garden in London, also almost always has some ambiguity as to its extent. In geography, location is considered to be more precise than "place". Relative location A relative location, or situation, is described as a displacement from another site. An example is "3 miles northwest of Seattle". Absolute lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptical Distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid, respectively, in iso-density plots. In statistics, the normal distribution is used in ''classical'' multivariate analysis, while elliptical distributions are used in ''generalized'' multivariate analysis, for the study of symmetric distributions with tails that are heavy, like the multivariate t-distribution, or light (in comparison with the normal distribution). Some statistical methods that were originally motivated by the study of the normal distribution have good performance for general elliptical distributions (with finite variance), particularly for spherical distributions (which are defined below). Elliptical distributions are also used in robust statistics to evaluate proposed multivari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Student's T-test
A ''t''-test is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of a Scale parameter, scaling term in the test statistic were known (typically, the scaling term is unknown and therefore a nuisance parameter). When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's ''t'' distribution. The ''t''-test's most common application is to test whether the means of two populations are different. History The term "''t''-statistic" is abbreviated from "hypothesis test statistic". In statistics, the t-distribution was first derived as a Posterior probability, posterior distribution in 1876 by Friedrich Robert Helmert, Helmert and Jacob Lüroth, Lüroth. The t-distribution also appeared in a more general form as Pearson Type Pearson distribution, IV di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \Re(z_1), \Re(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.Davis (1972) 6.2.2 p.258 A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac. A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function thatDavis (1972) 6.2.1 p.258 : \Beta(m,n) =\dfrac = \frac \B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta-prime Distribution
In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. Definitions Beta prime distribution is defined for x > 0 with two parameters ''α'' and ''β'', having the probability density function: : f(x) = \frac where ''B'' is the Beta function. The cumulative distribution function is : F(x; \alpha,\beta)=I_\left(\alpha, \beta \right) , where ''I'' is the regularized incomplete beta function. The expected value, variance, and other details of the distribution are given in the sidebox; for \beta>4, the excess kurtosis is :\gamma_2 = 6\frac. While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fisher-Snedecor Distribution
In probability theory and statistics, the ''F''-distribution or F-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA) and other ''F''-tests. Definition The F-distribution with ''d''1 and ''d''2 degrees of freedom is the distribution of : X = \frac where S_1 and S_2 are independent random variables with chi-square distributions with respective degrees of freedom d_1 and d_2. It can be shown to follow that the probability density function (pdf) for ''X'' is given by : \begin f(x; d_1,d_2) &= \frac \\ pt&=\frac \left(\frac\right)^ x^ \left(1+\frac \, x \right)^ \end for real ''x'' > 0. Here \mathrm is the beta function. In many applications, the parameters ''d''1 and ''d''2 are positive integers, but the distribution is well-def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Copula (statistics)
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval , 1 Copulas are used to describe/model the dependence (inter-correlation) between random variables. Their name, introduced by applied mathematician Abe Sklar in 1959, comes from the Latin for "link" or "tie", similar but unrelated to grammatical copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications. Sklar's theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables. Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulae ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mahalanobis Distance
The Mahalanobis distance is a measure of the distance between a point ''P'' and a distribution ''D'', introduced by P. C. Mahalanobis in 1936. Mahalanobis's definition was prompted by the problem of identifying the similarities of skulls based on measurements in 1927. It is a multi-dimensional generalization of the idea of measuring how many standard deviations away ''P'' is from the mean of ''D''. This distance is zero for ''P'' at the mean of ''D'' and grows as ''P'' moves away from the mean along each principal component axis. If each of these axes is re-scaled to have unit variance, then the Mahalanobis distance corresponds to standard Euclidean distance in the transformed space. The Mahalanobis distance is thus unitless, scale-invariant, and takes into account the correlations of the data set. Definition Given a probability distribution Q on \R^N, with mean \vec = (\mu_1, \mu_2, \mu_3, \dots , \mu_N)^\mathsf and positive-definite covariance matrix S, the Mahalanobis dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Complement
In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' × ''p'', and ''q'' × ''q'' matrices of complex numbers. Let :M = \left begin A & B \\ C & D \end\right/math> so that ''M'' is a (''p'' + ''q'') × (''p'' + ''q'') matrix. If ''D'' is invertible, then the Schur complement of the block ''D'' of the matrix ''M'' is the ''p'' × ''p'' matrix defined by :M/D := A - BD^C. If ''A'' is invertible, the Schur complement of the block ''A'' of the matrix ''M'' is the ''q'' × ''q'' matrix defined by :M/A := D - CA^B. In the case that ''A'' or ''D'' is singular, substituting a generalized inverse for the inverses on ''M/A'' and ''M/D'' yields the generalized Schur complement. The Schur complement is named after Issai Schur who used it to prove Schur's lemma, although it had been used previous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Integration
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals. There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, sequential Monte Carlo (also known as a particle filter), and mean-field particle methods. Overview In numerical integration, methods such as the trapezoidal rule use a deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
