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Generalized Gamma Distribution
The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models in survival analysis (such as the exponential distribution, the Weibull distribution and the gamma distribution) are special cases of the generalized gamma, it is sometimes used to determine which parametric model is appropriate for a given set of data. Another example is the half-normal distribution. Characteristics The generalized gamma distribution has two shape parameters, d > 0 and p > 0, and a scale parameter, a > 0. For non-negative ''x'' from a generalized gamma distribution, the probability density function is : f(x; a, d, p) = \frac, where \Gamma(\cdot) denotes the gamma function. The cumulative distribution function is : F(x; a, d, p) = \frac , where \gamma(\cdot) denotes the lowe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantile Function
In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability. Intuitively, the quantile function associates with a range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the percentile function, percent-point function or inverse cumulative distribution function. Definition Strictly monotonic distribution function With reference to a continuous and strictly monotonic cumulative distribution function F_X\colon \mathbb \to ,1/math> of a random variable ''X'', the quantile function Q\colon , 1\to \mathbb returns a threshold value ''x'' below which random draws from the given c.d.f. would fall ''100*p'' percent of the time. In terms of the distribution function ''F'', the qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modified Half-normal Distribution
In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero. Properties Using the \sigma parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by : f_Y(y; \sigma) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if \sigma is near zero), obtained by setting \theta=\frac, the probability density function is given by : f_Y(y; \theta) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. The cumulative distribution function (CDF) is given by : F_Y(y; \sigma) = \int_0^y \frac\sqrt \, \exp \left( -\frac \righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified Gaussian Distribution
In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated Gaussian distribution with interval (0,\infty)) as a result of censoring. Density function The probability density function of a rectified Gaussian distribution, for which random variables ''X'' having this distribution, derived from the normal distribution \mathcal(\mu,\sigma^2), are displayed as X \sim \mathcal^(\mu,\sigma^2) , is given by f(x;\mu,\sigma^2) =\Phi\delta(x)+ \frac\; e^\textrm(x). Here, \Phi(x) is the cumulative distribution function (cdf) of the standard normal distribution: \Phi(x) = \frac \int_^x e^ \, dt \quad x\in\mathbb, \delta(x) is the Dirac delta function \delta(x) = \begin +\infty, & x = 0 \\ 0, & x \ne 0 \end and, \textrm(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Folded Normal Distribution
The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal distribution. Such a case may be encountered if only the magnitude of some variable is recorded, but not its sign. The distribution is called "folded" because probability mass to the left of ''x'' = 0 is folded over by taking the absolute value. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. Definitions Density The probability density function (PDF) is given by :f_Y(x;\mu,\sigma^2)= \frac \, e^ + \frac \, e^ for ''x'' ≥ 0, and 0 everywhere else. An alternative formulation is given by : f\left(x \right)=\sqrte^\cosh, where cosh is the cosine Hyperbolic function. It foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Normal Distribution
In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics. Definitions Suppose X has a normal distribution with mean \mu and variance \sigma^2 and lies within the interval (a,b), \text \; -\infty \leq a < b \leq \infty . Then conditional on has a truncated normal distribution. Its , , for , is given by : and by otherwise. Here, : is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-t Distribution
In statistics, the folded-''t'' and half-''t'' distributions are derived from Student's ''t''-distribution by taking the absolute values of variates. This is analogous to the folded-normal and the half-normal statistical distributions being derived from the normal distribution. Definitions The folded non-standardized ''t'' distribution is the distribution of the absolute value of the non-standardized ''t'' distribution with \nu degrees of freedom; its probability density function is given by: :g\left(x\right)\;=\;\frac\left\lbrace \left +\frac\frac\right+\left +\frac\frac\right \right\rbrace \qquad(\mbox\quad x \geq 0). The half-''t'' distribution results as the special case of \mu=0, and the standardized version as the special case of \sigma=1. If \mu=0, the folded-''t'' distribution reduces to the special case of the half-''t'' distribution. Its probability density function then simplifies to :g\left(x\right)\;=\;\frac \left(1+\frac\frac\right)^ \qquad(\mbox\quad x \geq 0). Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
R (programming Language)
R is a programming language for statistical computing and graphics supported by the R Core Team and the R Foundation for Statistical Computing. Created by statisticians Ross Ihaka and Robert Gentleman, R is used among data miners, bioinformaticians and statisticians for data analysis and developing statistical software. Users have created packages to augment the functions of the R language. According to user surveys and studies of scholarly literature databases, R is one of the most commonly used programming languages used in data mining. R ranks 12th in the TIOBE index, a measure of programming language popularity, in which the language peaked in 8th place in August 2020. The official R software environment is an open-source free software environment within the GNU package, available under the GNU General Public License. It is written primarily in C, Fortran, and R itself (partially self-hosting). Precompiled executables are provided for various operating systems. R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strictly concave on (0,\infty). The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning double-gamma). Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the derivative with respect to gives: :\Gamma'(z+1)=z\Gamma'(z)+\Gamma(z) \, Dividing by or the equivalent gives: :\frac=\frac+\frac or: :\psi(z+1)=\psi(z)+\frac Since the harmonic numbers are defined for positive integers as :H_n=\sum_^n \frac 1 k, the digamma function is related to them by :\psi(n)=H_-\gamma, where and is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values : \psi \left(n+\tfrac12\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luigi Amoroso
Luigi Amoroso (26 March 1886 – 28 October 1965) was an Italian neoclassical economist influenced by Vilfredo Pareto. He provided support for and influenced the economic policy during the fascist regime. Work The microeconomical concept of the Amoroso–Robinson relation is named after him (and Joan Robinson): according to paper he is one of the first economists to have studied the dynamical equilibrium theory by using an analogy between economic systems and classical mechanics, thus applying to theories of economical behaviour mathematical tools as the calculus of variation. In his young years he contributed to the theory of functions of several complex variables, giving for the first time a set of necessary and sufficient conditions for the solvability of the Dirichlet problem for holomorphic functions of several variables in the paper . Also, in 1927 he provided to his former '' Normale'' schoolfellow Mauro Picone the funding for the creation of the '' Istituto Nazionale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nakagami Distribution
The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling spread \Omega>0. Characterization Its probability density function (pdf) is : f(x;\,m,\Omega) = \fracx^\exp\left(-\fracx^2\right), \forall x\geq 0. where (m\geq 1/2,\text\Omega>0) Its cumulative distribution function is : F(x;\,m,\Omega) = P\left(m, \fracx^2\right) where ''P'' is the regularized (lower) incomplete gamma function. Parametrization The parameters m and \Omega are : m = \frac , and : \Omega = \operatorname \left ^2 \right Parameter estimation An alternative way of fitting the distribution is to re-parametrize \Omega and ''m'' as ''σ'' = Ω/''m'' and ''m''. Given independent observations X_1=x_1,\ldots,X_n=x_n from the Nakagami distribution, the likelihood function is : L( \sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
