Natural exponential familyIn probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF). 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) where and are known functions.
Inverse transform samplingInverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the golden rule) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number between 0 and 1, interpreted as a probability, and then returns the smallest number such that for the cumulative distribution function of a random variable.
Survival analysisSurvival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology.
Infinite divisibility (probability)In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function. More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist n i.i.d. random variables Xn1, .
Wishart distributionIn statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy with GOE, GUE, GSE).
OverdispersionIn statistics, overdispersion is the presence of greater variability (statistical dispersion) in a data set than would be expected based on a given statistical model. A common task in applied statistics is choosing a parametric model to fit a given set of empirical observations. This necessitates an assessment of the fit of the chosen model. It is usually possible to choose the model parameters in such a way that the theoretical population mean of the model is approximately equal to the sample mean.
Rejection samplingIn 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 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.
Mixture distributionIn probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution.
Nakagami distributionThe 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 and a second parameter controlling spread . Its probability density function (pdf) is where Its cumulative distribution function is where P is the regularized (lower) incomplete gamma function. The parameters and are and An alternative way of fitting the distribution is to re-parametrize and m as σ = Ω/m and m.
Variance functionIn statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function is a measure of heteroscedasticity and plays a large role in many settings of statistical modelling. It is a main ingredient in the generalized linear model framework and a tool used in non-parametric regression, semiparametric regression and functional data analysis. In parametric modeling, variance functions take on a parametric form and explicitly describe the relationship between the variance and the mean of a random quantity.
