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Exponential Tilting
Exponential Tilting (ET), Exponential Twisting, or Exponential Change of Measure (ECM) is a distribution shifting technique used in many parts of mathematics. The different exponential tiltings of a random variable X is known as the natural exponential family of X. Exponential Tilting is used in Monte Carlo Estimation for rare-event simulation, and rejection and importance sampling in particular. In mathematical finance Exponential Tilting is also known as Esscher tilting (or the Esscher transform), and often combined with indirect Edgeworth approximation and is used in such contexts as insurance futures pricing. The earliest formalization of Exponential Tilting is often attributed to Esscher with its use in importance sampling being attributed to David Siegmund. Overview Given a random variable X with probability distribution \mathbb, density f, and moment generating function (MGF) M_X(\theta) = \mathbb ^. Example The exponentially tilted measure in many cases has the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Exponential Family
In probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF). Definition Univariate case The natural exponential families (NEF) are a subset of the exponential families. A NEF is an exponential family in which the natural parameter ''η'' and the natural statistic ''T''(''x'') are both the identity. A distribution in an exponential family with parameter ''θ'' can be written with probability density function (PDF) : f_X(x\mid \theta) = h(x)\ \exp\Big(\ \eta(\theta) T(x) - A(\theta)\ \Big) \,\! , where h(x) and A(\theta) are known functions. A distribution in a natural exponential family with parameter θ can thus be written with PDF : f_X(x\mid \theta) = h(x)\ \exp\Big(\ \theta x - A(\theta)\ \Big) \,\! . [Note that slightly different notation is used by the originator of the NEF, Carl Morris.Morris C. (2006) "Natural exponential families", ''Encyclopedia of Stat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the -th-order cumulant of their sum is equal to the sum of their -th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property. Just as for moments, where ''joint moments'' are used for collections of random variab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rejection Sampling
In numerical analysis and computational statistics, rejection sampling is a basic technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in \mathbb^m with a density. Rejection sampling is based on the observation that to sample a random variable in one dimension, one can perform a uniformly random sampling of the two-dimensional Cartesian graph, and keep the samples in the region under the graph of its density function. Note that this property can be extended to ''N''-dimension functions. Description To visualize the motivation behind rejection sampling, imagine graphing the density function of a random variable onto a large rectangular board and throwing darts at it. Assume that the darts are uniformly distributed around the board. Now remove all of the darts that are outside the area under the curve. The rem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Importance Sampling
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally attributed to a paper by Teun Kloek and Herman K. van Dijk in 1978, but its precursors can be found in statistical physics as early as 1949. Importance sampling is also related to umbrella sampling in computational physics. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both. Basic theory Let X\colon \Omega\to \mathbb be a random variable in some probability space (\Omega,\mathcal,P). We wish to estimate the expected value of ''X'' under ''P'', denoted E 'X;P'' If we have statistically independent random samples x_1, \ldots, x_n, generated according to ''P'', then an empirical estimate of E 'X;P''is : \widehat_ ;P= \frac \sum_^n x_i \quad \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruin Theory
In actuarial science and applied probability, ruin theory (sometimes risk theory or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin. Classical model The theoretical foundation of ruin theory, known as the Cramér–Lundberg model (or classical compound-Poisson risk model, classical risk process or Poisson risk process) was introduced in 1903 by the Swedish actuary Filip Lundberg. Lundberg's work was republished in the 1930s by Harald Cramér. The model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims. Premiums arrive a constant rate ''c'' > 0 from customers and claims arrive according to a Poisson process N_t with intensity ''λ'' and are independent and identically distributed non-negative rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G/G/1 Queue
In queueing theory, a discipline within the mathematical theory of probability, the G/G/1 queue represents the queue length in a system with a single server where interarrival times have a general (meaning arbitrary) distribution and service times have a (different) general distribution. The evolution of the queue can be described by the Lindley equation. The system is described in Kendall's notation where the ''G'' denotes a general distribution for both interarrival times and service times and the 1 that the model has a single server. Different interarrival and service times are considered to be independent, and sometimes the model is denoted GI/GI/1 to emphasise this. The numerical solution for the GI/G/1 can be obtained by discretizing the time. Waiting time Kingman's formula gives an approximation for the mean waiting time in a G/G/1 queue. Lindley's integral equation is a relationship satisfied by the stationary waiting time distribution which can be solved using the Wiener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations. Background Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stoc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale (probability Theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. History Originally, ''martingale'' referred to a class of betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users due to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem). This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddlepoint Approximation Method
The saddlepoint approximation method, initially proposed by Daniels (1954) is a specific example of the mathematical saddlepoint technique applied to statistics. It provides a highly accurate approximation formula for any PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ... or probability mass function of a distribution, based on the moment generating function. There is also a formula for the CDF of the distribution, proposed by Lugannani and Rice (1980). Definition If the moment generating function of a distribution is written as M(t) and the cumulant generating function as K(t) = \log(M(t)) then the saddlepoint approximation to the PDF of a distribution is defined as: :\hat(x) = \frac \exp(K(\hat) - \hatx) and the saddlepoint approximation to the CDF is defined as: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kullback–Leibler Divergence
In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how one probability distribution ''P'' is different from a second, reference probability distribution ''Q''. A simple interpretation of the KL divergence of ''P'' from ''Q'' is the expected excess surprise from using ''Q'' as a model when the actual distribution is ''P''. While it is a distance, it is not a metric, the most familiar type of distance: it is not symmetric in the two distributions (in contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence, a generalization of squared distance, and for certain classes of distributions (notably an exponential family), it satisfies a generalized Pythagorean theorem (which applies to squared distances). In the simple case, a relative entropy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
