|
Monotone Likelihood Ratio
A monotonic likelihood ratio in distributions f(x) and g(x) The ratio of the density functions above is increasing in the parameter x, so f(x)/g(x) satisfies the monotone likelihood ratio property. In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs). Formally, distributions ''ƒ''(''x'') and ''g''(''x'') bear the property if : \textx_1 > x_0, \quad \frac \geq \frac that is, if the ratio is nondecreasing in the argument x. If the functions are first-differentiable, the property may sometimes be stated :\frac \left( \frac \right) \geq 0 For two distributions that satisfy the definition with respect to some argument x, we say they "have the MLRP in ''x''." For a family of distributions that all satisfy the definition with respect to some statistic ''T''(''X''), we say they "have the MLR in ''T''(''X'')." Intuition The MLRP is used to represent a data-generating process that enjoys a straightforward re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" ( Herman Adèr quoting Kenneth Bollen). All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. Introduction Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event. As an example, consider a pair of ordinary six-sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanism Design
Mechanism design is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts at the end of the game, then goes backwards, it is also called reverse game theory. It has broad applications, from economics and politics in such fields as market design, auction theory and social choice theory to networked-systems (internet interdomain routing, sponsored search auctions). Mechanism design studies solution concepts for a class of private-information games. Leonid Hurwicz explains that 'in a design problem, the goal function is the main "given", while the mechanism is the unknown. Therefore, the design problem is the "inverse" of traditional economic theory, which is typically devoted to the analysis of the performance of a given mechanism.' So, two distinguishing features of these games are: * that a game "designer" choos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Dominance
Stochastic dominance is a partial order between random variables. It is a form of stochastic ordering. The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers. It is based on shared preferences regarding sets of possible outcomes and their associated probabilities. Only limited knowledge of preferences is required for determining dominance. Risk aversion is a factor only in second order stochastic dominance. Stochastic dominance does not give a total order, but rather only a partial order: for some pairs of gambles, neither one stochastically dominates the other, since different members of the broad class of decision-makers will differ regarding which gamble is preferable without them generally being considered to be equally attractive. Throughout the article, \rho, \nu stand for probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hazard Rate
Survival 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 Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ..., duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival? To answer such q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loss Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Minimum-variance Unbiased Estimator
In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter. For practical statistics problems, it is important to determine the MVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation. While combining the constraint of unbiasedness with the desirability metric of least variance leads to good results in most practical settings—making MVUE a natural starting point for a broad range of analyses—a targeted specification may perform better for a given problem; thus, MVUE is not always the best stopping point. Definition Consider estimation of g(\theta) based on data X_1, X_2, \ldots, X_n i.i.d. from some member of a family of densities p_\theta, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rao–Blackwell Theorem
In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria. The Rao–Blackwell theorem states that if ''g''(''X'') is any kind of estimator of a parameter θ, then the conditional expectation of ''g''(''X'') given ''T''(''X''), where ''T'' is a sufficient statistic, is typically a better estimator of θ, and is never worse. Sometimes one can very easily construct a very crude estimator ''g''(''X''), and then evaluate that conditional expected value to get an estimator that is in various senses optimal. The theorem is named after Calyampudi Radhakrishna Rao and David Blackwell. The process of transforming an estimator using the Rao–Blackwell theorem can be referred to as Rao–Blackwellization. The transformed estimator is called the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Statistics
The ''Annals of Statistics'' is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics. It was started in 1973 as a continuation in part of the '' Annals of Mathematical Statistics (1930)'', which was split into the ''Annals of Statistics'' and the ''Annals of Probability''. The journal CiteScore is 5.8, and its SCImago Journal Rank is 5.877, both from 2020. Articles older than 3 years are available on JSTOR, and all articles since 2004 are freely available on the arXiv. Editorial board The following persons have been editors of the journal: * Ingram Olkin (1972–1973) * I. Richard Savage (1974–1976) * Rupert Miller (1977–1979) * David V. Hinkley (1980–1982) * Michael D. Perlman (1983–1985) * Willem van Zwet (1986–1988) * Arthur Cohen (1988–1991) * Michael Woodroofe (1992–1994) * Larry Brown and John Rice (1995–1997) * Hans-Rudolf Künsch and James O. Berger (1998–2000) * John Marden and Jon A. Wellner (2001–2003) * M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Pfanzagl
Johann Richard Pfanzagl (2 July 1928 – 4 June 2019) was an Austrian mathematician known for his research in mathematical statistics. Life and career Pfanzagl studied from 1946 to 1951 at the University of Vienna and received his doctorate there in 1951 with Johann Radon and Edmund Hlawka on the topic of Hermitian forms in imaginary square number fields. In the same year he became a founding member of the Austrian Statistical Society, of which he was executive secretary from 1955 to 1959. From 1951 to 1959, Pfanzagl headed the statistical office of the Austrian Federal Economic Chamber. In 1959 he habilitated as a professor for statistics at the University of Vienna. Since 1960 he was a member of the Austrian Mathematical Society. At the same year, he moved to the University of Cologne, where he has held two chairs, one after the other, from 1960 to 1964 for economic and social statistics and from 1964 until his retirement in 1995 for mathematical statistics. Pfanzagl became ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Median-unbiased Estimator
In statistics and probability theory, the median is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the Arithmetic mean, mean (often simply described as the "average") is that it is not Skewness, skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result. Finite data set of numbers The median of a fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karlin–Rubin Theorem
In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 - \beta among all possible tests of a given size ''α''. For example, according to the Neyman–Pearson lemma, the likelihood-ratio test is UMP for testing simple (point) hypotheses. Setting Let X denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions f_(x), which depends on the unknown deterministic parameter \theta \in \Theta. The parameter space \Theta is partitioned into two disjoint sets \Theta_0 and \Theta_1. Let H_0 denote the hypothesis that \theta \in \Theta_0, and let H_1 denote the hypothesis that \theta \in \Theta_1. The binary test of hypotheses is performed using a test function \varphi(x) with a reject region R (a subset of measurement space). :\varphi(x) = \begin 1 & \text x \in R \\ 0 & \text x \in R^c \end meaning that H_1 is in force if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
