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Multivariate generalization of the gamma function
In mathematics, the multivariate gamma function Γp is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the
Multivariate_gamma_function
Extension of the factorial function
approximation Multiple gamma function Multivariate gamma function p-adic gamma function Pochhammer k-symbol Polygamma function q-gamma function Ramanujan's master
Gamma_function
Probability distribution on complex matrices
{\mathcal {C}}{\widetilde {\Gamma }}_{p}^{}(n)=\pi ^{p(p-1)/2}\prod _{j=1}^{p}\Gamma (n-j+1)} is the complex multivariate Gamma function. Using the trace rotation
Complex_Wishart_distribution
Generalization of gamma distribution to multiple dimensions
and Γp is the multivariate gamma function defined as Γ p ( n 2 ) = π p ( p − 1 ) / 4 ∏ j = 1 p Γ ( n 2 − j − 1 2 ) . {\displaystyle \Gamma _{p}\left({\frac
Wishart_distribution
function, Polygamma function Incomplete beta function Incomplete gamma function K-function Multivariate gamma function: A generalization of the Gamma
List of mathematical functions
List_of_mathematical_functions
Probability distribution
the determinant, and Γ p ( ⋅ ) {\displaystyle \Gamma _{p}(\cdot )} is the multivariate gamma function. If X ∼ W ( Σ , ν ) {\displaystyle {\mathbf {X}
Inverse-Wishart_distribution
Mathematical function
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
Beta_function
Probability distribution
function of a multivariate Cauchy distribution is given by: φ X ( t ) = e i x 0 ( t ) − γ ( t ) , {\displaystyle \varphi _{X}(t)=e^{ix_{0}(t)-\gamma (t)}
Cauchy_distribution
Multivariable generalization of the Student's t-distribution
In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization
Multivariate_t-distribution
Multivariate continuous probability distribution
| {\displaystyle |\cdot |} is the determinant, Γp(⋅) is the multivariate gamma function, and I p {\displaystyle {\textbf {I}}_{p}} is the p × p identity
Matrix_F-distribution
Two-parameter family of continuous probability distributions
scaled inverse chi-squared distribution. The inverse gamma distribution's probability density function is defined over the support x > 0 {\displaystyle x>0}
Inverse-gamma_distribution
Generalization of beta distribution
is the multivariate beta function: β p ( a , b ) = Γ p ( a ) Γ p ( b ) Γ p ( a + b ) {\displaystyle \beta _{p}\left(a,b\right)={\frac {\Gamma _{p}\left(a\right)\Gamma
Matrix variate beta distribution
Matrix_variate_beta_distribution
Concept in probability theory
terms of its characteristic function. The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution. It
Multivariate stable distribution
Multivariate_stable_distribution
Family of multivariate continuous probability distributions
statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions
Normal-inverse-gamma distribution
Normal-inverse-gamma_distribution
Fourier transform of the probability density function
characteristic functions generalizes to multivariate random variables and more complicated random elements. The argument of the characteristic function will always
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Probability distribution
{\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},} where γ ( α , β x ) {\displaystyle \gamma (\alpha ,\beta x)} is the lower incomplete gamma function. If
Gamma_distribution
probability theory and statistics, the generalized multivariate log-gamma (G-MVLG) distribution is a multivariate distribution introduced by Demirhan and Hamurkaroglu
Generalized multivariate log-gamma distribution
Generalized_multivariate_log-gamma_distribution
Mathematical functions having established names and notations
to Atle Selberg, the multivariate gamma function, and types of Bessel functions. The NIST Digital Library of Mathematical Functions has a section covering
Special_functions
Number of subsets of a given size
generalized to two real or complex valued arguments using the gamma function or beta function via ( x y ) = Γ ( x + 1 ) Γ ( y + 1 ) Γ ( x − y + 1 ) = 1 (
Binomial_coefficient
Formal power series
generating function in several variables can be generalized to arrays with multiple indices. These non-polynomial double sum examples are called multivariate generating
Generating_function
Generalization of gamma distribution
similarly, e.g. as the conjugate prior of the precision matrix of a multivariate normal distribution and matrix normal distribution. The compound distribution
Matrix_gamma_distribution
Probability distribution
similarly, e.g. as the conjugate prior of the covariance matrix of a multivariate normal distribution or matrix normal distribution. The compound distribution
Inverse matrix gamma distribution
Inverse_matrix_gamma_distribution
where C Γ p ( ν ) {\displaystyle {\mathcal {C}}\Gamma _{p}(\nu )} is the complex multivariate Gamma function C Γ p ( ν ) = π 1 2 p ( p − 1 ) ∏ j = 1 p Γ (
Complex inverse Wishart distribution
Complex_inverse_Wishart_distribution
Complex-differentiable (mathematical) function
{\displaystyle U} . Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function f {\displaystyle f} , this is equivalent
Holomorphic_function
theorem Multiplicities of entries in Pascal's triangle Multiset Multivariate gamma function Narayana numbers Negative binomial distribution Nörlund–Rice
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Probability distribution
{\displaystyle y_{i}} for all y i {\displaystyle y_{i}} . The multivariate generalized gamma (MGG) pdf can be derived from the MGB pdf by substituting b
Generalized_beta_distribution
Probability distribution
is the number of degrees of freedom, and Γ {\displaystyle \Gamma } is the gamma function. This may also be written as f ( t ) = 1 ν B ( 1 2 , ν 2 ) (
Student's_t-distribution
Concept in statistics
{\Sigma }}|^{-{\frac {p}{2}}}.} Here Γ p {\displaystyle \Gamma _{p}} is the multivariate gamma function. If X ∼ T n × p ( ν , M , Σ , Ω ) {\displaystyle \mathbf
Matrix_t-distribution
Special mathematical function defined as sin(x)/x
}\left(1-{\frac {x^{2}}{n^{2}}}\right)} and is related to the gamma function Γ(x), as well as to Gauss' Pi function, through Euler's reflection formula: sin ( π x
Sinc_function
Statistics function
{\displaystyle \gamma >0} . As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, the Q-function can be approximated
Q-function
{e^{-x+\pi i(m/2-n)}}{\Gamma (1+n-m/2)}}U(m/2-n,1+m,x).} The function was studied by Cunningham in the context of a multivariate generalisation of the
Cunningham_function
Probability distribution and special case of gamma distribution
incomplete gamma function and P ( s , t ) {\textstyle P(s,t)} is the regularized gamma function. In a special case of k = 2 {\displaystyle k=2} this function has
Chi-squared_distribution
Probability distribution
The normalizing constant is the multivariate beta function, which can be expressed in terms of the gamma function: B ( α ) = ∏ i = 1 K Γ ( α i ) Γ (
Dirichlet_distribution
Tool in multivariate statistical analysis
a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis
Matérn_covariance_function
Generalization of the one-dimensional normal distribution to higher dimensions
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization
Multivariate normal distribution
Multivariate_normal_distribution
Continuous probability distribution
{\displaystyle \gamma _{2}={\frac {-6\Gamma _{1}^{4}+12\Gamma _{1}^{2}\Gamma _{2}-3\Gamma _{2}^{2}-4\Gamma _{1}\Gamma _{3}+\Gamma _{4}}{[\Gamma _{2}-\Gamma _{1}^{2}]^{2}}}}
Weibull_distribution
Mathematical function with multiple real-number arguments
In mathematics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being
Function of several real variables
Function_of_several_real_variables
Multifractal function used in terrain modeling and simulation
"Weierstrass Function". MathWorld. Multifractal terrain generation paper on arXiv Fractal terrain for vehicle simulation Multivariate W-M function on ResearchGate
Weierstrass–Mandelbrot function
Weierstrass–Mandelbrot_function
Statistical distribution
density function is f X ( x ) = λ r Γ ( r ) e − λ x x r − 1 ( x > 0 ; λ , r > 0 ) {\displaystyle f_{X}^{}(x)={\frac {\lambda ^{r}}{\Gamma (r)}}\
Generalized integer gamma distribution
Generalized_integer_gamma_distribution
Probability distribution
variance σ2, a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normal-inverse-gamma distribution. Logically
Normal_distribution
Concept in probability theory
respectively, or to the multivariate normal distribution and multivariate t-distribution in the multivariate cases. In terms of the inverse gamma, β {\displaystyle
Conjugate_prior
Solution of a confluent hypergeometric equation
gamma function Laguerre polynomials Parabolic cylinder function (or Weber function) Poisson–Charlier function Toronto functions Whittaker functions Mκ
Confluent hypergeometric function
Confluent_hypergeometric_function
Function related to statistics and probability theory
which is calculated via Bayes' rule. The likelihood function, parameterized by a (possibly multivariate) parameter θ {\textstyle \theta } , is usually defined
Likelihood_function
In statistics, a multivariate Pareto distribution is a multivariate extension of a univariate Pareto distribution. There are several different types of
Multivariate Pareto distribution
Multivariate_Pareto_distribution
Probability distribution
typical characterization of the symmetric multivariate Laplace distribution has the characteristic function: φ ( t ; μ , Σ ) = exp ( i μ ′ t ) 1 + 1
Multivariate Laplace distribution
Multivariate_Laplace_distribution
Probability distribution
{y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 0<y<1;\gamma _{1},\gamma _{2}>0,} where B( ) is the beta function. If W = μ + σ ( Y
Pareto_distribution
Field of combinatorics using complex analysis
earliest work on multivariate generating functions started in the 1970s using probabilistic methods. Development of further multivariate techniques started
Analytic_combinatorics
Statistical function that defines the quantiles of a probability distribution
focusing increasing attention on methods based on quantile functions, as they work well with multivariate techniques based on either copula or quasi-Monte-Carlo
Quantile_function
Probability distribution
∈ ( 0 , 1 ] ∪ { 2 } {\displaystyle \beta \in (0,1]\cup \{2\}} . The multivariate generalized normal distribution, i.e. the product of n {\displaystyle
Generalized normal distribution
Generalized_normal_distribution
Mathematical function for the probability a given outcome occurs in an experiment
to the inverse of the covariance matrix of a multivariate normal distribution; generalization of the gamma distribution The cache language models and other
Probability_distribution
generalization of the beta negative binomial distribution. The generalized multivariate log-gamma distribution The Marshall–Olkin exponential distribution The
List of probability distributions
List_of_probability_distributions
Branch of discrete mathematics
combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic
Combinatorics
Matrix-variate probability distribution
n}}(X'dX)={\frac {2^{n}\pi ^{pn/2}}{\Gamma _{n}({\tfrac {1}{2}}p)}},} where Γ n {\displaystyle \Gamma _{n}} is the multivariate gamma function. The uniform distribution
Uniform distribution on a Stiefel manifold
Uniform_distribution_on_a_Stiefel_manifold
Distributions in probability theory
statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers
Dirichlet-multinomial distribution
Dirichlet-multinomial_distribution
Statistical distribution for dependence between random variables
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each
Copula_(statistics)
Family of probability distributions related to the normal distribution
first need to expand the part of the log-partition function that involves the multivariate gamma function: log Γ p ( a ) = log ( π p ( p − 1 ) 4 ∏ j =
Exponential_family
Type of mathematical function
density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any
Logarithmically concave function
Logarithmically_concave_function
Mathematical function
affine shape adaptation. Also see multivariate normal distribution. A more general formulation of a Gaussian function with a flat-top and Gaussian fall-off
Gaussian_function
Concept in Bayesian statistics
γ {\displaystyle \gamma } -Smallest Credible Sets ( γ {\displaystyle \gamma } -SCS) can easily be generalized to the multivariate case, and are bounded
Credible_interval
How many standard deviations apart from the mean an observed datum is
test-takers who received lower scores than students A and B. "For some multivariate techniques such as multidimensional scaling and cluster analysis, the
Standard_score
Statistical model
space), such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint
Gaussian_process
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Loss function in machine learning
loss function with γ = 2 {\displaystyle \gamma =2} , specifically L ( t , y ) = 4 ℓ 2 ( y ) {\displaystyle L(t,y)=4\ell _{2}(y)} . Multivariate adaptive
Hinge_loss
Probability distribution
Gamma \left({\frac {k}{2}}\right)}},&x\geq 0;\\0,&{\text{otherwise}}.\end{cases}}} where Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The
Chi_distribution
Probability distribution
developed in Chan and Tong (1986), which applies to multivariate cases beyond normality, e.g. skew multivariate t distribution and others. The distribution is
Skew_normal_distribution
Multivariate parameter family of continuous probability distributions
}}_{0})\right\}} Here Γ D [ ⋅ ] {\displaystyle \Gamma _{D}[\cdot ]} is the multivariate gamma function and T r ( Ψ ) {\displaystyle Tr({\boldsymbol {\Psi
Normal-inverse-Wishart distribution
Normal-inverse-Wishart_distribution
Approximation of a function by a polynomial
mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions. It provided the mathematical basis for some landmark early
Taylor's_theorem
method of moments Generalized multidimensional scaling Generalized multivariate log-gamma distribution Generalized normal distribution Generalized p-value
List_of_statistics_articles
Representation of a type of random process
{\begin{bmatrix}\gamma _{1}\\\gamma _{2}\\\gamma _{3}\\\vdots \\\gamma _{p}\\\end{bmatrix}}={\begin{bmatrix}\gamma _{0}&\gamma _{-1}&\gamma _{-2}&\cdots \\\gamma _{1}&\gamma
Autoregressive_model
Matrix of second derivatives
partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix
Hessian_matrix
Probability distribution
function route is favorable. If we define y ~ = − y {\displaystyle {\tilde {y}}=-y} then c ( y ~ ) {\displaystyle c({\tilde {y}})} above is a Gamma distribution
Distribution of the product of two random variables
Distribution_of_the_product_of_two_random_variables
Class of statistical estimators
}}_{n},{\hat {\gamma }}_{n}):=\mathop {\arg \max } _{\beta ,\gamma }\sum _{i=1}^{N}\displaystyle q(w_{i},\beta ,\gamma )} Assuming the function q is differentiable
M-estimator
Function equal to the product of its values on coprime factors
{\displaystyle \gamma (n)} , defined by γ ( n ) = ( − 1 ) ω ( n ) {\displaystyle \gamma (n)=(-1)^{\omega (n)}} , where the additive function ω ( n ) {\displaystyle
Multiplicative_function
Bound on probability of a random variable being far from its mean
{\kappa -\gamma ^{2}-1}{(\kappa -\gamma ^{2}-1)(1+k^{2})+(k^{2}-k\gamma -1)}}.} The necessity of k 2 − k γ − 1 > 0 {\displaystyle k^{2}-k\gamma -1>0} may
Chebyshev's_inequality
Branch of statistics
or multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution
Mathematical_statistics
Particular case of the generalized extreme value distribution
distributions. Theory related to the generalized multivariate log-gamma distribution provides a multivariate version of the Gumbel distribution. Gumbel has
Gumbel_distribution
Canadian statistician
under the supervision of Arak Mathai. Provost's research focuses on multivariate analysis, orthogonal series expansions, statistical modelling, complex
Serge_Provost_(statistician)
Probability distribution
-1}\end{aligned}}} where Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The beta function, B {\displaystyle \mathrm {B} } , is a normalization
Beta_distribution
Type of statistics
Γ , S ) = ( R , B ) {\displaystyle (\Gamma ,S)=(\mathbb {R} ,{\mathcal {B}})} , The empirical influence function is defined as follows. Let n ∈ N ∗ {\displaystyle
Robust_statistics
Integral of the Gaussian function, equal to sqrt(π)
t {\textstyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}dt} is the gamma function. More generally, ∫ 0 ∞ x n e − a x b d x = Γ ( ( n + 1 ) / b ) b a (
Gaussian_integral
Statistical relationship
variance) is only a sufficient statistic if the data is drawn from a multivariate normal distribution. As a result, the Pearson correlation coefficient
Correlation
Mathematical method in calculus
several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V. The product rule
Integration_by_parts
Probability distribution
distributions, such as the normal, binomial, gamma, and Poisson distributions. The probability density function (pdf) of an exponential distribution is f
Exponential_distribution
Set of probability distributions
family. In the multivariate case, the n-dimensional random variable X {\displaystyle \mathbf {X} } has a probability density function of the following
Exponential_dispersion_model
Probability distribution
{\displaystyle p(x\mid \nu ,{\hat {\mu }},{\hat {\sigma }})={\frac {\Gamma ({\frac {\nu +1}{2}})}{x\Gamma ({\frac {\nu }{2}}){\sqrt {\pi \nu }}{\hat {\sigma }}\,}}\left(1+{\frac
Log-t_distribution
Directed Transfer Function (DTF) and Partial Directed Coherence (PDC). These measures are defined in the framework of Multivariate Autoregressive Model
Brain_connectivity_estimators
Discrete probability distribution
using the lgamma function in the C standard library (C99 version) or R, the gammaln function in MATLAB or SciPy, or the log_gamma function in Fortran 2008
Poisson_distribution
Family of distributions that generalize the multivariate normal distribution
evaluate proposed multivariate-statistical procedures. Elliptical distributions are defined in terms of the characteristic function of probability theory
Elliptical_distribution
Type of data measuring one attribute
Univariate analysis can yield misleading results in cases in which multivariate analysis is more appropriate. Central tendency is one of the most common
Univariate_(statistics)
Probability distribution used in multivariate hypothesis testing
probability distribution used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test and multivariate analysis of variance (MANOVA)
Wilks's_lambda_distribution
Statistical distribution of complex random variables
{\begin{aligned}&\Gamma =V_{XX}+V_{YY}+i(V_{YX}-V_{XY}),\\&C=V_{XX}-V_{YY}+i(V_{YX}+V_{XY}).\end{aligned}}} The probability density function for complex normal
Complex_normal_distribution
Parametric model in survival analysis
partly as their cumulative distribution functions do not have a closed form. Finally, the generalized gamma distribution is a three-parameter distribution
Accelerated failure time model
Accelerated_failure_time_model
{\text{where}}\quad \beta ={\frac {\pi \xi ^{2}}{2\sigma ^{2}}}.\end{aligned}}} The multivariate generalization of the split normal distribution was proposed by Villani
Split_normal_distribution
Table that displays the frequency of variables
or crosstab) is a type of table in a matrix format that displays the multivariate frequency distribution of the variables. They are heavily used in survey
Contingency_table
Mathematical theorem
tools. In fact, Royen generalized the conjecture and proved it for multivariate gamma distributions. The proof did not gain attention when it was published
Gaussian correlation inequality
Gaussian_correlation_inequality
Class of probability distributions
binomial, negative binomial, normal, and gamma – are a special subset of NEF, called NEF with quadratic variance function (NEF-QVF) because the variance can
Natural_exponential_family
Smooth function in statistics
for Normal, Bernoulli, Poisson, and Gamma. In addition, we describe the applications and use of variance functions in maximum likelihood estimation and
Variance_function
Measure of variation in statistics
the gamma function, and equals: c 4 ( N ) = 2 N − 1 Γ ( N 2 ) Γ ( N − 1 2 ) . {\displaystyle c_{4}(N)\,=\,{\sqrt {\frac {2}{N-1}}}\,\,\,{\frac {\Gamma {\left({\frac
Standard_deviation
Correlation inequality
chain coupling argument. The lattice condition for μ is also called multivariate total positivity, and sometimes the strong FKG condition; the term (multiplicative)
FKG_inequality
Class of statistical models
canonical link functions and their inverses (sometimes referred to as the mean function, as done here). In the cases of the exponential and gamma distributions
Generalized_linear_model
MULTIVARIATE GAMMA-FUNCTION
MULTIVARIATE GAMMA-FUNCTION
MULTIVARIATE GAMMA-FUNCTION
MULTIVARIATE GAMMA-FUNCTION
MULTIVARIATE GAMMA-FUNCTION
MULTIVARIATE GAMMA-FUNCTION
MULTIVARIATE GAMMA-FUNCTION
MULTIVARIATE GAMMA-FUNCTION
MULTIVARIATE GAMMA-FUNCTION