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Deficiency (statistics)
In statistics, the deficiency is a measure to compare a statistical model with another statistical model. The concept was introduced in the 1960s by the French mathematician Lucien Le Cam, who used it to prove an approximative version of the Blackwell–Sherman–Stein theorem. Closely related is the Le Cam distance, a Pseudometric space, pseudometric for the maximum deficiency between two statistical models. If the deficiency of a model \mathcal in relation to \mathcal is zero, then one says \mathcal is ''better'' or ''more informative'' or ''stronger'' than \mathcal. Introduction Le Cam defined the statistical model more abstract than a probability space with a family of probability measures. He also didn't use the term "statistical model" and instead used the term "experiment". In his publication from 1964 he introduced the statistical experiment to a parameter set \Theta as a triple (X,E,(P_\theta)_) consisting of a set X, a vector lattice E with unit I and a family of normal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model represents, often in considerably idealized form, the Data generating process, data-generating process. When referring specifically to probability, probabilities, the corresponding term is probabilistic model. All Statistical hypothesis testing, statistical hypothesis tests and all Estimator, statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. 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 J. Adèr, Herman Adèr quoting Kenneth A. Bollen, Kenneth Bollen). Introduction Informally, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucien Le Cam
Lucien Marie Le Cam (November 18, 1924 – April 25, 2000) was a mathematician and statistician. Biography Le Cam was born November 18, 1924, in Croze, France. His parents were farmers, and unable to afford higher education for him; his father died when he was 13. After graduating from a Catholic school in 1942, he began studying at a seminary in Limoges, but immediately quit upon learning that he would not be allowed to study chemistry there. Instead he continued his studies at a lycée, which did not teach chemistry but did teach mathematics. In May 1944 he joined an underground group, and then went into hiding, returning to his school the following November but soon afterwards moving to Paris, where he began studying at the University of Paris. He graduated in 1945 with the degree '' licence ès sciences''.. Le Cam then worked for a hydroelectric utility for five years, while meeting at the University of Paris for a weekly seminar in statistics. In 1950, he was invited to beco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematical Statistics
The ''Annals of Mathematical Statistics'' was a peer-reviewed statistics journal published by the Institute of Mathematical Statistics from 1930 to 1972. It was superseded by the '' Annals of Statistics'' and the '' Annals of Probability''. In 1938, Samuel Wilks became editor-in-chief of the ''Annals'' and recruited a remarkable editorial staff: Fisher, Neyman, Cramér, Hotelling, Egon Pearson Egon Sharpe Pearson (11 August 1895 – 12 June 1980) was one of three children of Karl Pearson and Maria, née Sharpe, and, like his father, a British statistician. Career Pearson was educated at Winchester College and Trinity College ..., Georges Darmois, Allen T. Craig, Deming, von Mises, H. L. Rietz, and Shewhart. References External links ''Annals of Mathematical Statistics''at Project Euclid Statistics journals Probability journals {{statistics-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Institute Of Mathematical Statistics
The Institute of Mathematical Statistics is an international professional and scholarly society devoted to the development, dissemination, and application of statistics and probability. The Institute currently has about 4,000 members in all parts of the world. Beginning in 2005, the institute started offering joint membership with the Bernoulli Society for Mathematical Statistics and Probability as well as with the International Statistical Institute. The Institute was founded in 1935 with Harry C. Carver and Henry L. Rietz as its two most important supporters. The institute publishes a variety of journals, and holds several international conference every year. Publications The Institute publishes five journals: *'' Annals of Statistics'' *'' Annals of Applied Statistics'' *'' Annals of Probability'' *'' Annals of Applied Probability'' *''Statistical Science'' In addition, it co-sponsors: * '' Electronic Communications in Probability'' * '' Electronic Journal of Probability'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudometric Space
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy, the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis. When a topology is generated using a family of pseudometrics, the space is called a gauge space. Definition A pseudometric space (X,d) is a set X together with a non-negative real-valued function d : X \times X \longrightarrow \R_, called a , such that for every x, y, z \in X, #d(x,x) = 0. #''Symmetry'': d(x,y) = d(y,x) #'' Subadditivity''/''Triangle inequality'': d(x,z) \leq d(x,y) + d(y,z) Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have d(x, y) = 0 for dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Lattice
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Sur la décomposition des opérations fonctionelles linéaires''. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis. Definition Preliminaries If X is an ordered vector space (which by definition is a vector space over the reals) and if S is a subset of X then an element b \in X is an upper bound (resp. lower bound) of S if s \leq b (resp. s \geq b) for all s \in S. An eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Functional
In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, \leq) is a linear functional f on V so that for all positive elements v \in V, that is v \geq 0, it holds that f(v) \geq 0. In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When V is a complex vector space, it is assumed that for all v\ge0, f(v) is real. As in the case when V is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace W\subseteq V, and the partial order does not extend to all of V, in which case the positive elements of V are the positive elements of W, by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any x \in V equal to s^s for some s \in V to a real num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract L-space
In mathematics, specifically in order theory and functional analysis, an abstract ''L''-space, an AL-space, or an abstract Lebesgue space is a Banach lattice (X, \, \cdot \, ) whose norm is additive on the positive cone of ''X''. In probability theory, it means the standard probability space. Examples The strong dual of an AM-space with unit is an AL-space. Properties The reason for the name abstract ''L''-space is because every AL-space is isomorphic (as a Banach lattice) with some subspace of L^1(\mu). Every AL-space ''X'' is an order complete vector lattice of minimal type; however, the order dual of ''X'', denoted by ''X''+, is ''not'' of minimal type unless ''X'' is finite-dimensional. Each order interval in an AL-space is weakly compact. The strong dual of an AL-space is an AM-space with unit. The continuous dual space X^ (which is equal to ''X''+) of an AL-space ''X'' is a Banach lattice that can be identified with C_ ( K ), where ''K'' is a compact extre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Lattice
In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order, \leq, such that for all , the implication \Rightarrow holds, where the absolute value is defined as , x, = x \vee -x := \sup\\text Examples and constructions Banach lattices are extremely common in functional analysis, and "every known example n 1948of a Banach space asalso a vector lattice." In particular: * , together with its absolute value as a norm, is a Banach lattice. * Let be a topological space, a Banach lattice and the space of continuous bounded functions from to with norm \, f\, _ = \sup_ \, f(x)\, _Y\text Then is a Banach lattice under the pointwise partial order: \Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text Examples of non-lattice Banach spaces are now known; James' space is one such.Kania, Tomasz (12 April 2017).Answerto "Banach space that is not a Banach lattice" (accessed 13 August 2022). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
