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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 and special case of gamma distribution
chi-squared distribution. The following is a special case of Cochran's theorem. Theorem. If Z 1 , . . . , Z n {\displaystyle Z_{1},...,Z_{n}} are independent
Chi-squared_distribution
Surname list
Forfarshire. The Cochrans are traditionally mainly a Western Lowlands family. Notable people with the surname include: Alexander Gilmore Cochran (1846–1928)
Cochran
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 t-distribution
Normal_distribution
with multiplicity) is b. Theorem: For any non-trivial PBD, v ≤ b. This result also generalizes the Erdős–De Bruijn theorem: For a PBD with λ = 1 having
Fisher's_inequality
Degradation of AI models trained on synthetic data
explicitly construct them in terms of independent random variables using Cochran's theorem. To be precise, μ 1 {\displaystyle \mu _{1}} and σ 1 {\displaystyle
Model_collapse
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
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
Correction for sample variance bias
Var ( X 1 ) {\displaystyle =n(n-1)\operatorname {Var} (X_{1})} Cochran's theorem Bias of an estimator Standard deviation Unbiased estimation of standard
Bessel's_correction
Continuous probability distribution
{\displaystyle U_{2}} (defined above) might be demonstrated by applying Cochran's theorem. Equivalently, since the chi-squared distribution is the sum of squares
F-distribution
Statistical approach
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Response_surface_methodology
Probability distribution
distribution with ν = n − 1 {\displaystyle \nu =n-1} degrees of freedom (by Cochran's theorem). It is readily shown that the quantity Z = ( X ¯ n − μ ) n σ {\displaystyle
Student's_t-distribution
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Bose–Mesner_algebra
Statistical measure of how far values spread from their average
that Yi are independent observations from a normal distribution, Cochran's theorem shows that the unbiased sample variance S2 follows a scaled chi-squared
Variance
Probability distribution
chi-squared distributions. Cramér's theorem Cochran's theorem Infinite divisibility (probability) Khinchin's theorem on the factorization of distributions
Indecomposable_distribution
Collection of statistical models
Rosenbaum (2002, page 40) cites Section 5.7 (Permutation Tests), Theorem 2.3 (actually Theorem 3, page 184) of Lehmann's Testing Statistical Hypotheses (1959)
Analysis_of_variance
Family of distributions that generalize the multivariate normal distribution
good properties. Under finite-variance assumptions, an extension of Cochran's theorem (on the distribution of quadratic forms) holds. An elliptical distribution
Elliptical_distribution
Geometric relation between a triangle's side lengths and cevian length
In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the
Stewart's_theorem
British-American statistician
Cochran (15 July 1909 – 29 March 1980) was a prominent statistician. He was born in Scotland but spent most of his life in the United States. Cochran
William_Gemmell_Cochran
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Generalized randomized block design
Generalized_randomized_block_design
Procedure to estimate standard deviation from a sample
To derive the correction, note that for normally distributed X, Cochran's theorem implies that ( n − 1 ) s 2 / σ 2 {\displaystyle (n-1)s^{2}/\sigma
Unbiased estimation of standard deviation
Unbiased_estimation_of_standard_deviation
Design of tasks
with some reservations. In 1950, Gertrude Mary Cox and William Gemmell Cochran published the book Experimental Designs, which became the major reference
Design_of_experiments
General linear model that blends ANOVA and regression
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Analysis_of_covariance
Bias in causal inference
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Confounding
Experimental setup in behavioral sciences
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Paradigm_(experimental)
Strategy) Coalescent theory Cochran's C test Cochran's Q test Cochran's theorem Cochran–Armitage test for trend Cochran–Mantel–Haenszel statistics Cochrane–Orcutt
List_of_statistics_articles
Cameron, RC Bose, JA … - 1973 - US Dept. of Commerce, National … Some Theorems for Partially Balanced Designs, W. S. Connor and W. H. Clatworthy, Ann
W._H._Clatworthy
American econometrician (1915-2003)
intractable for N > 9. The next day, he received a note from Cochran asking him to try out Cochran's theorem, which turned out to be the answer. In 1962, Anderson
Richard_Loree_Anderson
Mathematical problem
MacNeish's theorem does not give a very good lower bound, for instance if n ≡ 2 (mod 4), that is, there is a single 2 in the prime factorization, the theorem gives
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Probability distribution
sample mean when forming the sample covariance, a consequence of Cochran's theorem. Similarly, V T Σ − 1 V V T S − 1 V ∼ χ ν − p + 1 2 , {\displaystyle
Ratio_distribution
Experimental design that is optimal with respect to some statistical criterion
of mean-unbiased estimators (under the conditions of the Gauss–Markov theorem). In the estimation theory for statistical models with one real parameter
Optimal_experimental_design
Design of experiments to collect similar contexts together
be considered almost independent. The blocks method helps proving limit theorems in the case of dependent random variables. The blocks method was introduced
Blocking_(statistics)
Experimental design framework
{\displaystyle \xi } , the posterior probability can be calculated using Bayes' theorem p ( θ ∣ y , ξ ) = p ( y ∣ θ , ξ ) p ( θ ) p ( y ∣ ξ ) , {\displaystyle
Bayesian_experimental_design
Cancer epidemiologist and biostatistician
Maryland in 1953. His thesis advisor was William G. Cochran a statistician known for Cochran's theorem, Cochran-Mantel-Haenzel Test and author of standard biostatistical
Calvin_Zippin
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
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Scheirer–Ray–Hare_test
Theory in statistics
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Association_scheme
Statistical theorem
In statistics, Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produce confidence intervals
Wilks'_theorem
French mathematician (1843–1913)
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Gaston_Tarry
Interpretation of probability
sequential use of Bayes' theorem: as more data become available, calculate the posterior distribution using Bayes' theorem; subsequently, the posterior
Bayesian_probability
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Robust_parameter_design
British statistician and geneticist (1919–2000)
Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Oscar_Kempthorne
Statistical principle
on an assumption of the distributional form (see Pitman–Koopman–Darmois theorem below), but remained very important in theoretical work. Roughly, given
Sufficient_statistic
Technique in integral evaluation
Glasser's master theorem Pushforward measure Swokowski 1983, p. 257 Swokowski 1983, p. 258 Briggs & Cochran 2011, p. 361 Rudin 1987, Theorem 7.26 Spivak 1965
Integration_by_substitution
Theoretically optimal hypothesis test
1-\beta (\theta )=\operatorname {E} [\varphi (X)|\theta ].} The Karlin–Rubin theorem (named for Samuel Karlin and Herman Rubin) can be regarded as an extension
Uniformly_most_powerful_test
Theorem of stationary processes
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
Wold's_theorem
Test used in the analysis of stratified or matched categorical data
In statistics, the Cochran–Mantel–Haenszel test (CMH) is a test used in the analysis of stratified or matched categorical data. It allows an investigator
Cochran–Mantel–Haenszel statistics
Cochran–Mantel–Haenszel_statistics
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
Statistical test comparing two probability distributions
two distribution functions across all x values. By the Glivenko–Cantelli theorem, if the sample comes from the distribution F(x), then Dn converges to 0
Kolmogorov–Smirnov_test
Method of statistical inference
/ˈbeɪʒən/ BAY-zhən) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence
Bayesian_inference
Shape with three equal sides
may be derived from the formula of an isosceles triangle by Pythagoras theorem: the altitude h {\displaystyle h} of a triangle is the square root of the
Equilateral_triangle
Conditional probability used in Bayesian statistics
this student is a girl? The correct answer can be computed using Bayes' theorem. The event G is that the student observed is a girl, and the event T is
Posterior_probability
American mathematician
1215/S0012-7094-07-13723-2. S2CID 119495376. Cochran, Tim D.; Harvey, Shelly (2008). "Homology and Derived Series of Groups II: Dwyer's Theorem". Geometry and Topology. 12
Tim_Cochran
Correlation of a signal with a time-shifted copy of itself, as a function of shift
{\displaystyle 0} for all other τ {\displaystyle \tau } . The Wiener–Khinchin theorem relates the autocorrelation function R X X {\displaystyle \operatorname
Autocorrelation
Shape with three sides
An important tool for proving the existence of these points is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent
Triangle
Branch of mathematics
of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained
Geometry
Statistics term
statistic which is not complete. This is important because the Lehmann–Scheffé theorem cannot be applied to such models. Galili and Meilijson 2016 propose the
Completeness_(statistics)
Function related to statistics and probability theory
{\text{HH}})=0.25} , a conclusion which could only be reached via Bayes' theorem given knowledge about the marginal probabilities P ( p H = 0.5 ) {\textstyle
Likelihood_function
Statistical test that compares goodness of fit
embedded in. Multiplying by −2 ensures mathematically that (by Wilks' theorem) λ LR {\displaystyle \lambda _{\text{LR}}} converges asymptotically to
Likelihood-ratio_test
Family of probability distributions related to the normal distribution
distributed data using a fixed number of values. (Pitman–Koopman–Darmois theorem) Exponential families have conjugate priors, an important property in Bayesian
Exponential_family
Statistical property
variance needs to be computed according to the Markov chain central limit theorem. There are cases when a sample is taken without knowing, in advance, how
Standard_error
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
Type of average of a collection of numbers
radius r = AQ = AG. Using Pythagoras' theorem, QM² = AQ² + AM² ∴ QM = √AQ² + AM² = QM. Using Pythagoras' theorem, AM² = AG² + GM² ∴ GM = √AM² − AG² = GM
Arithmetic_mean
Statistical test
determine, making the t-test more convenient. Because of the central limit theorem, many test statistics are approximately normally distributed for large
Z-test
American mathematician
American Mathematical Society. Cochran, Tim D.; Harvey, Shelly (2008), "Homology and derived series of groups. II. Dwyer's theorem", Geometry & Topology, 12
Shelly_Harvey
Statistical distribution for dependence between random variables
and minimize tail risk and portfolio-optimization applications. Sklar's theorem states that any multivariate joint distribution can be written in terms
Copula_(statistics)
Approximation method in statistics
after reading Gauss's work, Laplace, after proving the central limit theorem, used it to give a large sample justification for the method of least squares
Least_squares
Statistic measuring inter-rater agreement for categorical items
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Cohen's_kappa
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Coefficient_of_variation
Measure of linear correlation
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Pearson correlation coefficient
Pearson_correlation_coefficient
Mathematical concept
Wolf 1998, p. 208, Theorem 7.2 Smith, Eggen & St. Andre 2006, pg. 141 Theorem 3.3(a) Lay 2006, p. 71, Theorem 7.26 Briggs & Cochran 2011, pp. 28–29 Lay
Inverse_function
N-th root of the product of n numbers
radius r = AQ = AG. Using Pythagoras' theorem, QM² = AQ² + AM² ∴ QM = √AQ² + AM² = QM. Using Pythagoras' theorem, AM² = AG² + GM² ∴ GM = √AM² − AG² = GM
Geometric_mean
Measure of the joint variability
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Covariance
Range to estimate an unknown parameter
Two widely applicable methods are bootstrapping and the central limit theorem. The latter method works only if the sample is large, since it entails
Confidence_interval
Distribution function associated with the empirical measure of a sample
1 to that underlying distribution, according to the Glivenko–Cantelli theorem. A number of results exist to quantify the rate of convergence of the empirical
Empirical distribution function
Empirical_distribution_function
Statistical hypothesis test
{\displaystyle {\bar {x}}} is assumed to be normal. By the central limit theorem, if the observations are independent and the second moment exists, then
Student's_t-test
Unbiased statistical estimator minimizing variance
can prove there is an essentially unique MVUE. Using the Rao–Blackwell theorem one can also prove that determining the MVUE is simply a matter of finding
Minimum-variance unbiased estimator
Minimum-variance_unbiased_estimator
Statistical method
analytical form or an asymptotic theory (e.g., an applicable central limit theorem) to help estimate the distribution of the statistics of interest. This
Bootstrapping_(statistics)
Mathematical function for the probability a given outcome occurs in an experiment
is uncountable or countable, respectively. The Lebesgue decomposition theorem states that any probability distribution on the real line can be uniquely
Probability_distribution
Method of estimating the parameters of a statistical model, given observations
a proof published by Samuel S. Wilks in 1938, now called Wilks' theorem. The theorem shows that the error in the logarithm of likelihood values for estimates
Maximum_likelihood_estimation
Statistical hypothesis test
of the 2 × 1 chi-squared test for goodness of fit, see binomial test. Cochran–Mantel–Haenszel chi-squared test. McNemar's test, used in certain 2 × 2
Chi-squared_test
Processes that maintain quality at a constant level
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Quality_control
Empirical law on the variance of species in a habitat
Tweedie distribution. A limit theorem for independent and identically distributed variables, as with the Tweedie convergence theorem, might then be considered
Taylor's_law
Subfield of information theory and computer science
algorithmic information. Instead of proving similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily
Algorithmic information theory
Algorithmic_information_theory
Data visualization
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Box_plot
Variance outlier test
Cochran's C {\displaystyle C} test, named after William G. Cochran, is a one-sided upper limit variance outlier statistical test . The C test is used to
Cochran's_C_test
How many standard deviations apart from the mean an observed datum is
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Standard_score
Study of convergence properties of statistical estimators
θ 0 {\displaystyle \theta _{0}} , then it is consistent (by Slutsky's theorem). If ( θ ^ n ) n ∈ N {\displaystyle ({\hat {\theta }}_{n})_{n\in \mathbb
Asymptotic theory (statistics)
Asymptotic_theory_(statistics)
Statistical phenomenon
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Regression_toward_the_mean
Diagnostic plot of binary classifier ability
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Receiver operating characteristic
Receiver_operating_characteristic
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
Branch of statistics
moment estimator is also asymptotically normal (due to the central limit theorem and the delta method). Least square estimation (LSE): This method applies
Parametric_statistics
Experiment methodology
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
A/B_testing
Measure of distance between two proportions
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Cohen's_h
Generates a forecast of future values of a time series
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Exponential_smoothing
Psychometric measurement scale
defensible approximation to an interval scale, in which case the central limit theorem allows treatment of the data as interval data measuring a latent variable
Likert_scale
Statistical model for a binary dependent variable
Regression validation Mean and predicted response Errors and residuals Goodness of fit Studentized residual Gauss–Markov theorem Mathematics portal v t e
Logistic_regression
Covariance and correlation
g\right)=\left(f\star f\right)\star \left(g\star g\right)} . Analogous to the convolution theorem, the cross-correlation satisfies F { f ⋆ g } = F { f } ¯ ⋅ F { g } , {\displaystyle
Cross-correlation
Measure of statistical dispersion
Bertil., Westergren (1988). Beta [beta] mathematics handbook : concepts, theorems, methods, algorithms, formulas, graphs, tables. Studentlitteratur. p. 348
Interquartile_range
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