# Category: Theorems in statistics

Binomial sum variance inequality
The binomial sum variance inequality states that the variance of the sum of binomially distributed random variables will always be less than or equal to the variance of a binomial variable with the sa
Law of total variance
In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if and are random
Optional stopping theorem
In probability theory, the optional stopping theorem (or Doob's optional sampling theorem) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its in
Welch–Satterthwaite equation
In statistics and uncertainty analysis, the Welch–Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom of a linear combination of independent sample varianc
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions
Slutsky's theorem
In probability theory, Slutsky’s theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. The theorem was named after Eugen Slu
Stein's lemma
Stein's lemma, named in honor of Charles Stein, is a theorem of probability theory that is of interest primarily because of its applications to statistical inference — in particular, to James–Stein es
Bernstein–von Mises theorem
In Bayesian inference, the Bernstein-von Mises theorem provides the basis for using Bayesian credible sets for confidence statements in parametric models. It states that under some conditions, a poste
Bruck–Ryser–Chowla theorem
The Bruck–Ryser–Chowla theorem is a result on the combinatorics of block designs that implies nonexistence of certain kinds of design. It states that if a (v, b, r, k, λ)-design exists with v = b (a s
Wold's theorem
In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wo
Basu's theorem
In statistics, Basu's theorem states that any boundedly complete minimal sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata Basu. It is often used in st
Schuette–Nesbitt formula
In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after and Cecil J. Nesbitt. The probabilistic version of the Schuette–Nesbitt formula
Gauss–Markov theorem
In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear un
Craps principle
In probability theory, the craps principle is a theorem about event probabilities under repeated iid trials. Let and denote two mutually exclusive events which might occur on a given trial. Then the p
Fieller's theorem
In statistics, Fieller's theorem allows the calculation of a confidence interval for the ratio of two means.
Itô's lemma
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent fun
Law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results
Law of the iterated logarithm
In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khin
Central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distri
Raikov's theorem
Raikov’s theorem, named for Russian mathematician Dmitrii Abramovich Raikov, is a result in probability theory. It is well known that if each of two independent random variables ξ1 and ξ2 has a Poisso
Lyapunov's central limit theorem
No description available.
Law of total cumulance
In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total varia
Cox's theorem
Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" in
In mathematics, Varadhan's lemma is a result from large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic φ(Zε) of a fa
Bapat–Beg theorem
In probability theory, the Bapat–Beg theorem gives the joint probability distribution of order statistics of independent but not necessarily identically distributed random variables in terms of the cu
Bochner's theorem
In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's th
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 es
Neyman–Pearson lemma
In statistics, the Neyman–Pearson lemma was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933. The Neyman-Pearson lemma is part of the Neyman-Pearson theory of statistical testing, which
Glivenko–Cantelli theorem
In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determ
Le Cam's theorem
In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following. Suppose: * are independent random variables, each with a Bernoulli distribution (i.e., equal to
Robbins lemma
In statistics, the Robbins lemma, named after Herbert Robbins, states that if X is a random variable having a Poisson distribution with parameter λ, and f is any function for which the expected value
Frisch–Waugh–Lovell theorem
In econometrics, the Frisch–Waugh–Lovell (FWL) theorem is named after the econometricians Ragnar Frisch, Frederick V. Waugh, and Michael C. Lovell. The Frisch–Waugh–Lovell theorem states that if the r
Lindeberg's condition
In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independe
Shannon–Hartley theorem
In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is
Cramér's decomposition theorem
Cramér’s decomposition theorem for a normal distribution is a result of probability theory. It is well known that, given independent normally distributed random variables ξ1, ξ2, their sum is normally
Freidlin–Wentzell theorem
In mathematics, the Freidlin–Wentzell theorem (due to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentze
Law of total covariance
In probability theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are random variables on the same probability space, a
Aumann's agreement theorem
Aumann's agreement theorem was stated and proved by Robert Aumann in a paper titled "Agreeing to Disagree", which introduced the set theoretic description of common knowledge. The theorem concerns age
Lehmann–Scheffé theorem
In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation. The theorem states that any estima
Skorokhod's representation theorem
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can
Asymptotic equipartition property
In information theory, the asymptotic equipartition property (AEP) is a general property of the output samples of a stochastic source. It is fundamental to the concept of typical set used in theories
Hammersley–Clifford theorem
The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics that gives necessary and sufficient conditions under which a strictly positive prob
Cochran's theorem
In statistics, Cochran's theorem, devised by William G. Cochran, is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance.
Pyrrho's lemma
In statistics, Pyrrho's lemma is the result that if one adds just one extra variable as a regressor from a suitable set to a linear regression model, one can get any desired outcome in terms of the co
Continuous mapping theorem
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s defi
Lévy's continuity theorem
In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variable
Donsker's theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the centra
Fisher–Tippett–Gnedenko theorem
In statistics, the Fisher–Tippett–Gnedenko theorem (also the Fisher–Tippett theorem or the extreme value theorem) is a general result in extreme value theory regarding asymptotic distribution of extre
Kosambi–Karhunen–Loève theorem
In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem is a representation of a stochastic pro
Doob–Meyer decomposition theorem
The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasi
Characterization of probability distributions
In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc. – is the only one that possesses properties specified in the theorem. A characterization
Kac–Bernstein theorem
The Kac–Bernstein theorem is one of the first characterization theorems of mathematical statistics. It is easy to see that if the random variables and are independent and normally distributed with the
Hájek–Le Cam convolution theorem
In statistics, the Hájek–Le Cam convolution theorem states that any in a parametric model is asymptotically equivalent to a sum of two independent random variables, one of which is normal with asympto
Berry–Esseen theorem
In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the