Lévy distributionIn probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile. It is a special case of the inverse-gamma distribution. It is a stable distribution. The probability density function of the Lévy distribution over the domain is where is the location parameter and is the scale parameter.
Maximum likelihood estimationIn statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.
Financial risk modelingFinancial risk modeling is the use of formal mathematical and econometric techniques to measure, monitor and control the market risk, credit risk, and operational risk on a firm's balance sheet, on a bank's trading book, or re a fund manager's portfolio value; see Financial risk management. Risk modeling is one of many subtasks within the broader area of financial modeling. Risk modeling uses a variety of techniques including market risk, value at risk (VaR), historical simulation (HS), or extreme value theory (EVT) in order to analyze a portfolio and make forecasts of the likely losses that would be incurred for a variety of risks.
Mathematical statisticsMathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory. Statistical data collection is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling.
Uncertainty quantificationUncertainty quantification (UQ) is the science of quantitative characterization and estimation of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known. An example would be to predict the acceleration of a human body in a head-on crash with another car: even if the speed was exactly known, small differences in the manufacturing of individual cars, how tightly every bolt has been tightened, etc.
Unbiased estimation of standard deviationIn statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the use of significance tests and confidence intervals, or by using Bayesian analysis.
Rational expectationsRational expectations is an economic theory used to explain how individuals make predictions about the future based on all available information. It states that individuals will also learn from past trends and experiences in order to make the best possible prediction about what will happen. They could be wrong sometimes, but that, on average, they will be correct. The concept of rational expectations was first introduced by John F. Muth in his paper "Rational Expectations and the Theory of Price Movements" published in 1961.
Sensitivity analysisSensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagation of uncertainty; ideally, uncertainty and sensitivity analysis should be run in tandem.
Robust measures of scaleIn statistics, robust measures of scale are methods that quantify the statistical dispersion in a sample of numerical data while resisting outliers. The most common such robust statistics are the interquartile range (IQR) and the median absolute deviation (MAD). These are contrasted with conventional or non-robust measures of scale, such as sample standard deviation, which are greatly influenced by outliers.
Structural equation modelingStructural equation modeling (SEM) is a diverse set of methods used by scientists doing both observational and experimental research. SEM is used mostly in the social and behavioral sciences but it is also used in epidemiology, business, and other fields. A definition of SEM is difficult without reference to technical language, but a good starting place is the name itself. SEM involves a model representing how various aspects of some phenomenon are thought to causally connect to one another.
Logistic distributionIn probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). The logistic distribution is a special case of the Tukey lambda distribution.
Joint probability distributionGiven two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s).
