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Theorem in statistics
In statistics, Basu's theorem states that any boundedly complete and sufficient statistic is independent of any ancillary statistic. This is a 1955 result
Basu's_theorem
Surname list
'Bright one'. Basus are considered as Kulin Kayasthas of Gautam gotra, along with Ghoshes, Mitras and Bangaja (Eastern Bengal) Guhas. Amrita Basu, (b. 1953)
Basu
American statistician and academic (1924–2014)
special case of "Basu's theorem", a few years before the publication by Deb Basu. Hogg's second paper on the topic of Basu's theorem was never published
Robert_V._Hogg
Indian statistician
frequentist statistics; Basu's paradoxes were especially important in the development of survey sampling. In statistical theory, Basu's theorem established the
Debabrata_Basu
Statistics term
minimum-variance unbiased estimator. Bounded completeness occurs in Basu's theorem, which states that a statistic that is both boundedly complete and sufficient
Completeness_(statistics)
Statistic whose sampling distribution does not depend on the parameter
construct prediction intervals. They are also used in connection with Basu's theorem to prove independence between statistics. This concept was first introduced
Ancillary_statistic
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
Statistical theorem in the analysis of variance
In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics
Cochran's_theorem
Probability distribution
sample standard deviation, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's
Normal_distribution
ionizations in context of stellar atmospheres. Basu's theorem: The Basu's theorem, a result of Debabrata Basu (1955) states that any complete sufficient statistic
Timeline_of_Indian_innovation
Statistical theorem
In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result that characterizes the transformation
Rao–Blackwell_theorem
Every polynomial has a real or complex root
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Indian economist and academic (born 1952)
published a paper giving a new proof of the Pythagoras theorem, via a property of isosceles triangles. Basu has held visiting professorships at the Massachusetts
Kaushik_Basu
Indian inventions
what else has been similarly overlooked". Basu's theorem – The Basu's theorem, a result of Debabrata Basu (1955) states that any complete sufficient
List of Indian inventions and discoveries
List_of_Indian_inventions_and_discoveries
Theorem in statistics
Lehmann–Scheffé theorem provides sufficient conditions for the existence of a best unbiased estimator in a statistical model. The theorem states that any
Lehmann–Scheffé_theorem
The polynomial hierarchy is contained in probabilistic Turing machine in polynomial time
machines) was proved by Saugata Basu and Thierry Zell in 2010 and a complex analogue of Toda's theorem was proved by Saugata Basu in 2011. The proof is broken
Toda's_theorem
Statistics concept
sample mean can be shown to be independent of each other, using, e.g. Basu's theorem. That fact, and the normal and chi-squared distributions given above
Errors_and_residuals
Physics theorem
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete
Virial_theorem
Theorem in statistics and econometrics
econometrics, the Frisch–Waugh–Lovell (FWL) theorem proves a property of ordinary least squares estimators. The theorem is named for econometricians Ragnar Frisch
Frisch–Waugh–Lovell_theorem
Statistical principle
Completeness of a statistic Basu's theorem on independence of complete sufficient and ancillary statistics Lehmann–Scheffé theorem: a complete sufficient estimator
Sufficient_statistic
interpolation Bartlett's method Bartlett's test Bartlett's theorem Base rate Baseball statistics Basu's theorem Bates distribution Baum–Welch algorithm Bayes classifier
List_of_statistics_articles
Counting polynomial roots in an interval
derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of p located in an interval
Sturm's_theorem
Fellow of the Royal College of Surgeons of England Debabrata Basu, founded Basu's theorem Anil Kumar Bhattacharya, known for Bhattacharya coefficient,
List_of_Bengalis
Study of rational collective decision-making
impossibility theorem is what often comes to mind when one thinks about impossibility theorems in voting. There are several famous theorems concerning social
Social_choice_theory
Power series with rational exponents
Newton (1736) cf. Kedlaya (2001), introduction Basu &al (2006), chapter 2 ("Real Closed Fields"), theorem 2.91 (p. 75) Cherlin (1976), chapter 2 ("The Ax–Kochen–Ershof
Puiseux_series
Field in mathematics similar to the real numbers
fields of hyperreal numbers that include infinitesimals. In algebra, most theorems that involve the real numbers remain true when formulated for arbitrary
Real_closed_field
exponential polynomial on G. Ritt's theorem states that the analogues of unique factorization and the factor theorem hold for the ring of exponential polynomials
Exponential_polynomial
Analysis of datasets using techniques from topology
first classification theorem for persistent homology appeared in 1994 via Barannikov's canonical forms. The classification theorem interpreting persistence
Topological_data_analysis
Statement in arithmetic combinatorics
In arithmetic combinatorics, the corners theorem states that for every ε > 0 {\displaystyle \varepsilon >0} , for large enough N {\displaystyle N} , any
Corners_theorem
German polymath and scholar (1777–1855)
Waltershausen 1856, p. 100. Basu, Soham; Velleman, Daniel J. (21 April 2017). "On Gauss's first proof of the fundamental theorem of algebra". arXiv:1704.06585
Carl_Friedrich_Gauss
Indian mathematician (1887–1920)
notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything
Srinivasa_Ramanujan
Non-crossing graph with vertices on outer face
drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors K4 and K2,3, or by their
Outerplanar_graph
Topics referred to by the same term
that created the Universe Penrose–Hawking singularity theorems, in general relativity theory, theorems about how gravitation produces singularities such as
Singularity
Algebraic surface defined by a cubic polynomial
(2024) Hartshorne (1997), Theorem V.4.11. Kollár, Smith, Corti (2004), Exercise 1.29. Kollár, Smith, Corti (2004), Theorems 1.37 and 1.38. Kollár, Smith
Cubic_surface
Study of systems of inequalitites
set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem. Related fields are o-minimal theory and real analytic geometry. Examples:
Real_algebraic_geometry
Feature of systems that defy description
generalized Kolmogorov complexity. Instead of proving similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily
Complexity
Branch of mathematics
theory. Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach. In classical algebraic geometry
Algebraic_geometry
Universal code which encodes positive integers into binary code words
Universal code Varicode, a practical application Zeckendorf's theorem Maximal entropy random walk Basu, Manjusri; Prasad, Bandhu (2010-09-01). "Long range variations
Fibonacci_coding
Country in South Asia
BCE) contain the earliest extant verbal expression of the Pythagorean theorem (although very likely it had been known to the Old Babylonians.) All mathematical
India
American mathematician
types and polytopes, and a generalization of the Hadwiger transversal theorem to higher dimensions. He and Goodman were the founding editors of the journal
Richard_M._Pollack
Concept in modern economics
of cooperation without coercion." Kaushik Basu has called the First Welfare Theorem the Invisible Hand Theorem. Some economists question the integrity of
Invisible_hand
which peaks at that interval. An application of Gravelle et al. welfare theorem is exemplified in market failure. Since 1998, Gravelle has published enormously
Hugh_Gravelle
Concept in physics
Non-equilibrium thermodynamics Stochastic thermodynamics Fluctuation theorem Linear response theory Maes, Christian (2020-03-27). "Frenesy: Time-symmetric
Frenesy_(physics)
Generalizations of the Riemann zeta function
{\displaystyle \Lambda } , the result follows. For k = 3 {\displaystyle k=3} , the theorem says ∑ σ ∈ Σ 3 S ( i σ ( 1 ) , i σ ( 2 ) , i σ ( 3 ) ) = ζ ( i 1 ) ζ (
Multiple_zeta_function
Process of using data analysis for predicting population data from sample data
about [estimators] based on very large samples, where the central limit theorem ensures that these [estimators] will have distributions that are nearly
Statistical_inference
1902 crime detective novel by Arthur Conan Doyle
The Dynamics of an Asteroid Reichenbach Falls A Treatise on the Binomial Theorem Studies Sherlockian game Holmesian studies The New Annotated Sherlock Holmes
The_Hound_of_the_Baskervilles
Flat-sided three-dimensional shape
results on polyhedral concepts, like Hilbert's third problem, Steinitz's theorem, and stellation of Platonic solids. Polyhedra are used in many fields,
Polyhedron
Hypothesis in Marxist economics
7, 1961, p. 92. Shalom Groll & Ze'ev Orzech, "From Marx to the Okishio theorem: a genealogy". History of Political Economy, Vol. 21, Issue 2, 1989, pp
Tendency of the rate of profit to fall
Tendency_of_the_rate_of_profit_to_fall
Economics concept
members thanks to his altruism model, also known as the Rotten kid theorem. The theorem asserts that one person, sometimes referred to as a benevolent dictator
Household_economics
British philosopher and economist (born 1947)
these dimensions are linked by what he calls the interpersonal addition theorem, which supports the utilitarian principle of distribution. In his book
John_Broome_(philosopher)
Mathematical problem
called "finding the chromatic number of the plane". By the de Bruijn–Erdős theorem, a result of de Bruijn & Erdős (1951), the problem is equivalent (under
Hadwiger–Nelson_problem
Indian theoretical chemist
Generalized Wick's theorem had been the methods of directly determining the various reduced density matrices via generalized Brillouin's theorem and the contracted
Debashis_Mukherjee
Type of matrix
distances are used to avoid computing square roots and to simplify relevant theorems and algorithms. Euclidean distance matrices are closely related to Gram
Euclidean_distance_matrix
American physicist (born 1955)
in the same direction, they could not do so (this is the Mermin-Wagner theorem). However, they can all walk in the same direction. The Toner–Tu equations
John_Toner_(physicist)
Middle quantile of a data set or probability distribution
This concept is relevant to voting theory on account of the median voter theorem. When it exists, the median in all directions coincides with the geometric
Median
Concept in statistics
representation theorem by Bruno de Finetti (later extended by other probability theorists such as Halmos and Savage). The extended versions of the theorem show
Exchangeable_random_variables
Standardized AI performance test
Xueguang; Xu, Jianyu; Wang, Xinyi; Xia, Tony (December 2023). "TheoremQA: A Theorem-driven Question Answering Dataset". In Bouamor, Houda; Pino, Juan;
Language_model_benchmark
Intelligence of machines
Nilsson (1998, chpt. 3.3) Universal approximation theorem: Russell & Norvig (2021, p. 752) The theorem: Cybenko (1988), Hornik, Stinchcombe & White (1989)
Artificial_intelligence
Indian-American mathematician (1920–2023)
test, Rao's Quadratic Entropy, Cramér–Rao bound and the Rao–Blackwell theorem these are related to the quality of estimators. Majority of the publications
C._R._Rao
Decomposing n-space into cells in which each of a set of polynomials has constant sign
complexity than that resulting from the original proof of Tarski–Seidenberg theorem. It is efficient enough to be implemented on a computer. It is one of the
Cylindrical algebraic decomposition
Cylindrical_algebraic_decomposition
Roots of multiple multivariate polynomials
the solutions (in the case of a single equation, this is Abel–Ruffini theorem). The Barth surface, shown in the figure is the geometric representation
System of polynomial equations
System_of_polynomial_equations
American economist (born 1960)
cash, bonds, and stocks is inconsistent with the mutual-fund separation theorem, which states that all investors should hold the same composition of risky
David_N._Weil
Chemical element with atomic number 118 (Og)
elements, and this happens rarely. But look at how many equations and theorems in mathematics are named after somebody. And in medicine? Alzheimer, Parkinson
Oganesson
for number theory and algebraic geometry (Ramanujam–Samuel theorem, Ramanujam vanishing theorem), a mathematics student of Father Charles Racine Natesan
List of Loyola College, Chennai people
List_of_Loyola_College,_Chennai_people
Computational model used in machine learning
universal function approximator, as proven by the universal approximation theorem. However, the proof does not specify the number of neurons required, the
Neural network (machine learning)
Neural_network_(machine_learning)
Type of chart
Statistics Seeing Data. "Making sense of data visualizations". Seeing Data. Ron Basu (2004). Implementing Quality: A Practical Guide to Tools and Techniques.
Radar_chart
2006 Italian association football scandal
associate. It was necessary to ascertain the truth, not to frame people in a theorem that is the child of the Pirelli files." About Berlusconi, Moggi said:
Calciopoli
British polymath (1890–1962)
given age to the future growth of the population. Fisher's fundamental theorem of natural selection, which states that "the rate of increase in fitness
Ronald_Fisher
(ŒdO) Wang Bing China, France, United States, United Kingdom Marguerite's Theorem Le Théorème de Marguerite Anna Novion France, Switzerland Occupied City
2023_Cannes_Film_Festival
Method of quality control
(eds.). Current Issues in Statistical Inference: Essays in Honor of D. Basu. Hayward, CA: Institute of Mathematical Statistics. pp. 99–112. ISBN 978-0-940600-24-9
Statistical_process_control
proofs into formal proofs in order to automatically verify them. Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated
Applications of artificial intelligence
Applications_of_artificial_intelligence
existence of the luminiferous aether and the failures of the equipartition theorem in predicting the heat capacity of solids. Both problems were solved within
List of unsolved problems in physics
List_of_unsolved_problems_in_physics
Indian mathematician (1932–2020)
mathematician M. S. Narasimhan, for their proof of the Narasimhan–Seshadri theorem which proved the necessary conditions for stable vector bundles on a Riemann
C._S._Seshadri
Value that appears most often in a set of data
Statistica Neerlandica. 33 (1): 1–5. doi:10.1111/j.1467-9574.1979.tb00657.x. Basu, Sanjib; Dasgupta, Anirban (1997). "The mean, median, and mode of unimodal
Mode_(statistics)
Greatest common divisor of polynomials
Euclidean division of integers. Its existence is based on the following theorem: Given two univariate polynomials a {\textstyle a} and b ≠ 0 {\displaystyle
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
British economist
2307/2231407. JSTOR 2231407. Roberts, Kevin W.S. (January 1980). "Possibility theorems with interpersonally comparable welfare levels". The Review of Economic
Kevin_W._S._Roberts
Non-zero-sum game thought experiment
prisoner's dilemma. The original game scenario was formulated in 1994 by Kaushik Basu and goes as follows: "An airline loses two suitcases belonging to two different
Traveler's_dilemma
Approach to public-key cryptography
F q ) {\displaystyle E(\mathbb {F} _{q})} , it follows from Lagrange's theorem that the number h = 1 n | E ( F q ) | {\displaystyle h={\frac {1}{n}}|E(\mathbb
Elliptic-curve_cryptography
symbol for zero, interpolation formula, Fibonacci's identity, Brahmagupta's theorem, the first complete arithmetic solution (including zero and negative solutions)
Indian_people
224] Birth 2: Operation: Retrieve the Genius Dog! – Part 3 Fermat's Last Theorem (天才犬奪還作戦!3, Tensai Ken Dakkan Sakusen! San) [Chapter 225] Birth 2: Operation:
List_of_GetBackers_chapters
Indian mathematician (born 1951)
includes his famous work with Koblitz, now commonly called the Balu-Koblitz Theorem. His work in Additive Combinatorics includes his two page paper on additive
Ramachandran_Balasubramanian
Aspect of relativity in physics
zero quadrupole moment) will not radiate, in agreement with Birkhoff's theorem. More technically, the second time derivative of the quadrupole moment
Gravitational_wave
Economic theory that asset prices fully reflect all available information
random walk theory can be described through the fundamental theorem of asset pricing. This theorem provides mathematical predictions regarding the price of
Efficient-market_hypothesis
Property of having a unique mode or maximum value
normally distributed demands" (PDF). Method in appendix D, Example in theorem 2 page 5. Retrieved 2013-08-28. "Mathematical Programming Glossary". Retrieved
Unimodality
Inequality in information theory
inequality Csiszár, Imre; Körner, János (2011). Information Theory: Coding Theorems for Discrete Memoryless Systems. Cambridge University Press. p. 44. ISBN 9781139499989
Pinsker's_inequality
American economist and Nobel Laureate (born 1943)
population distributions are optimal). Stiglitz dubbed this the 'Henry George theorem' in reference to the radical classical economist Henry George who famously
Joseph_Stiglitz
Semi-classical method for computing quantum eigenvalues
Albert (1917). "Zum Quantensatz von Sommerfeld und Epstein" [On the Quantum Theorem of Sommerfeld and Epstein]. Deutsche Physikalische Gesellschaft, Verhandlungen
Einstein–Brillouin–Keller method
Einstein–Brillouin–Keller_method
1776 economics book by Adam Smith
training, trust, hardship, and unemployment). He also describes Smith's theorem that "the division of labour is limited by the extent of the market" as
The_Wealth_of_Nations
Iterative method for finding maximum likelihood estimates in statistical models
parameters θ(t), the conditional distribution of the Zi is determined by Bayes' theorem to be the proportional height of the normal density weighted by τ: T j
Expectation–maximization algorithm
Expectation–maximization_algorithm
and rules for summing series, Brahmagupta's identity, and Brahmagupta's theorem. Brahmagupta described gravity as a force of attraction. He posited that
Timeline_of_Indian_history
econometrics Anil Nerode (b. 1932), mathematician, proved the Myhill-Nerode Theorem Ria Persad (b. 1974), mathematician, classical musician, and model K. C
List_of_Indian_Americans
Discrete model of computation
fundamental result is the characterization in the Curtis–Hedlund–Lyndon theorem of the set of global rules of cellular automata as the set of continuous
Cellular_automaton
Continuous band of stars that appears on plots of stellar color versus brightness
mass and radius can be deduced. This became known as the Vogt–Russell theorem; named after Heinrich Vogt and Henry Norris Russell. It was subsequently
Main_sequence
Type of statistics
normally distributed, at least approximately, or that the central limit theorem can be relied on to produce normally distributed estimates. Unfortunately
Robust_statistics
Indian polymath (1907–1966)
series expressions for stochastic processes via the Kosambi–Karhunen–Loève theorem. He is also well known for his work in numismatics and for compiling critical
Damodar_Dharmananda_Kosambi
Polish-born English economist, known for the development of the Rybczynski theorem Anthony Saunders, chairman, Department of Finance, Stern School of Business
List of people associated with the London School of Economics
List_of_people_associated_with_the_London_School_of_Economics
Indian mathematical statistician (1930-1999)
1016/S0378-3758(00)00159-2. Laha, R. G. (1998). "An extension of the Darmois–Skitovich theorem". Math. Methods Statist. 7 (4): 479–483. Kagan A, Laha RG, Rohatgi V (1997)
Radha_Laha
Star at the centre of the Solar System
centre, releasing gravitational potential energy. According to the virial theorem, half of this released gravitational energy goes into heating, which leads
Sun
important for his work in pure mathematics, having authored a number of theorems. Frank Wilczek (born 1951): American theoretical physicist. Along with
List_of_agnostics
Swedish logician, philosopher, and mathematical statistician
119–170. doi:10.2307/2986734. JSTOR 2986734. Martin-Löf, P. The continuity theorem on a locally compact group. Teor. Verojatnost. i Primenen. 10 1965 367–371
Per_Martin-Löf
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