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Charles Stein (statistician)
Charles Max Stein (March 22, 1920 – November 24, 2016) was an American mathematical statistician and professor of statistics at Stanford University. He received his Ph.D in 1947 at Columbia University with advisor Abraham Wald. He held faculty positions at Berkeley and the University of Chicago before moving permanently to Stanford in 1953. He is known for Stein's paradox in decision theory, which shows that ordinary least squares estimates can be uniformly improved when many parameters are estimated; for Stein's lemma, giving a formula for the covariance of one random variable with the value of a function of another when the two random variables are jointly normally distributed; and for Stein's method, a way of proving theorems such as the Central Limit Theorem that does not require the variables to be independent and identically distributed. He was a member of the National Academy of Sciences. He died in November 2016 at the age of 96. Works *''Approximate Computat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Normal Distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be ''k''-variate normally distributed if every linear combination of its ''k'' components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Definitions Notation and parameterization The multivariate normal distribution of a ''k''-dimensional random vector \mathbf = (X_1,\ldots,X_k)^ can be written in the following notation: : \mathbf\ \sim\ \mathcal(\boldsymbol\mu,\, \boldsymbol\Sigma), or to make it explicitly known that ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Members Of The United States National Academy Of Sciences
Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in a database ** Member variable, a variable that is associated with a specific object * Limb (anatomy), an appendage of the human or animal body ** Euphemism for penis * Structural component of a truss, connected by nodes * User (computing), a person making use of a computing service, especially on the Internet * Member (geology), a component of a geological formation * Member of parliament A member of parliament (MP) is the representative in parliament of the people who live in their electoral district. In many countries with Bicameralism, bicameral parliaments, this term refers only to members of the lower house since upper house ... * The Members, a British punk rock band * Meronymy, a semantic relationship in linguistics * Church mem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Columbia University Alumni
Columbia may refer to: * Columbia (personification), the historical female national personification of the United States, and a poetic name for America Places North America Natural features * Columbia Plateau, a geologic and geographic region in the U.S. Pacific Northwest * Columbia River, in Canada and the United States ** Columbia Bar, a sandbar in the estuary of the Columbia River ** Columbia Country, the region of British Columbia encompassing the northern portion of that river's upper reaches *** Columbia Valley, a region within the Columbia Country ** Columbia Lake, a lake at the head of the Columbia River *** Columbia Wetlands, a protected area near Columbia Lake ** Columbia Slough, along the Columbia watercourse near Portland, Oregon * Glacial Lake Columbia, a proglacial lake in Washington state * Columbia Icefield, in the Canadian Rockies * Columbia Island (District of Columbia), in the Potomac River * Columbia Island (New York), in Long Island Sound Populated pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematicians
American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, people who self-identify their ancestry as "American" ** American English, the set of varieties of the English language native to the United States ** Native Americans in the United States, indigenous peoples of the United States * American, something of, from, or related to the Americas, also known as "America" ** Indigenous peoples of the Americas * American (word), for analysis and history of the meanings in various contexts Organizations * American Airlines, U.S.-based airline headquartered in Fort Worth, Texas * American Athletic Conference, an American college athletic conference * American Recordings (record label), a record label previously known as Def American * American University, in Washington, D.C. Sports teams Soccer * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2016 Deaths
This is a list of deaths of notable people, organised by year. New deaths articles are added to their respective month (e.g., Deaths in ) and then linked here. 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 1988 1987 See also * Lists of deaths by day * Deaths by year {{DEFAULTSORT:deaths by year ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1920 Births
Nineteen or 19 may refer to: * 19 (number), the natural number following 18 and preceding 20 * one of the years 19 BC, AD 19, 1919, 2019 Films * ''19'' (film), a 2001 Japanese film * ''Nineteen'' (film), a 1987 science fiction film Music * 19 (band) 19 was a Japanese pop/folk duo. Its members were Kenji Okahira and Keigo Iwase The Japanese language has a system of honorific speech, referred to as , parts of speech that show respect. Their use is mandatory in many social situations. Ho ..., a Japanese pop music duo Albums * ''19'' (Adele album), 2008 * ''19'', a 2003 album by Alsou * ''19'', a 2006 album by Evan Yo * ''19'', a 2018 album by MHD * ''19'', one half of the double album '' 63/19'' by Kool A.D. * '' Number Nineteen'', a 1971 album by American jazz pianist Mal Waldron * ''XIX'' (EP), a 2019 EP by 1the9 Songs * "19" (song), a 1985 song by British musician Paul Hardcastle. * "Nineteen", a song by Bad4Good from the 1992 album ''Refugee (Bad4 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stein Discrepancy
A Stein discrepancy is a statistical divergence between two probability measures that is rooted in Stein's method. It was first formulated as a tool to assess the quality of Markov chain Monte Carlo samplers,J. Gorham and L. Mackey. Measuring Sample Quality with Stein's Method. Advances in Neural Information Processing Systems, 2015. but has since been used in diverse settings in statistics, machine learning and computer science. Definition Let \mathcal be a measurable space and let \mathcal be a set of measurable functions of the form m : \mathcal \rightarrow \mathbb. A natural notion of distance between two probability distributions P, Q, defined on \mathcal, is provided by an integral probability metric : (1.1) \quad d_(P , Q) := \sup_ , \mathbb_(X)- \mathbb_(Y) , where for the purposes of exposition we assume that the expectations exist, and that the set \mathcal is sufficiently rich that (1.1) is indeed a metric on the set of probability distributions on \mathcal, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stein's Unbiased Risk Estimate
In statistics, Stein's unbiased risk estimate (SURE) is an unbiased estimator of the mean-squared error of "a nearly arbitrary, nonlinear biased estimator." In other words, it provides an indication of the accuracy of a given estimator. This is important since the true mean-squared error of an estimator is a function of the unknown parameter to be estimated, and thus cannot be determined exactly. The technique is named after its discoverer, Charles Stein. Formal statement Let \mu \in ^d be an unknown parameter and let x \in ^d be a measurement vector whose components are independent and distributed normally with mean \mu_i, i=1,...,d, and variance \sigma^2. Suppose h(x) is an estimator of \mu from x, and can be written h(x) = x + g(x), where g is weakly differentiable. Then, Stein's unbiased risk estimate is given by :\operatorname(h) = d\sigma^2 + \, g(x)\, ^2 + 2 \sigma^2 \sum_^d \frac g_i(x) = -d\sigma^2 + \, g(x)\, ^2 + 2 \sigma^2 \sum_^d \frac h_i(x), where g_i(x) is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James–Stein Estimator
The James–Stein estimator is a biased estimator of the mean, \boldsymbol\theta, of (possibly) correlated Gaussian distributed random vectors Y = \ with unknown means \. It arose sequentially in two main published papers, the earlier version of the estimator was developed by Charles Stein in 1956, which reached a relatively shocking conclusion that while the then usual estimate of the mean, or the sample mean written by Stein and James as (Y_i) = , is admissible when m \leq 2, however it is inadmissible when m \geq 3 and proposed a possible improvement to the estimator that shrinks the sample means towards a more central mean vector \boldsymbol\nu (which can be chosen a priori or commonly the "average of averages" of the sample means given all samples share the same size), is commonly referred to as Stein's example or paradox. This earlier result was improved later by Willard James and Charles Stein in 1961 through simplifying the original process. It can be shown that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Science
''Statistical Science'' is a review journal published by the Institute of Mathematical Statistics. The founding editor was Morris H. DeGroot, who explained the mission of the journal in his 1986 editorial: "A central purpose of ''Statistical Science'' is to convey the richness, breadth and unity of the field by presenting the full range of contemporary statistical thought at a modest technical level accessible to the wide community of practitioners, teachers, researchers and students of statistics and probability." Editors * 2017-2019 Cun-Hui Zhang * 2014-2016 Peter Green * 2011-2013 Jon Wellner * 2008-2010 David Madigan * 2005-2007 Ed George * 2002-2004 George Casella * 2001 Morris Eaton * 2001 Richard Tweedie * 1998-2000 Leon Gleser * 1995-1997 Paul Switzer * 1992-1994 Robert E. Kass The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
