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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
Statistical test that compares goodness of fit
they're embedded in. Multiplying by −2 ensures mathematically that (by Wilks' theorem) λ LR {\displaystyle \lambda _{\text{LR}}} converges asymptotically
Likelihood-ratio_test
American mathematician (1906–1964)
has been dubbed Wilks's theorem. Another result, also called “Wilks' theorem” occurs in the theory of likelihood ratio tests, where Wilks showed the distribution
Samuel_S._Wilks
Function related to statistics and probability theory
the log-likelihood ratio, considered as a test statistic, is given by Wilks' theorem. The likelihood ratio is also of central importance in Bayesian inference
Likelihood_function
Method of estimating the parameters of a statistical model, given observations
justification in a proof published by Samuel S. Wilks in 1938, now called Wilks' theorem. The theorem shows that the error in the logarithm of likelihood
Maximum_likelihood_estimation
Statistical test of variance
very difficult to determine. A convenient result, attributed to Samuel S. Wilks, says that as the sample size n approaches the test statistic has asymptotically
Omnibus_test
Test of normality in frequentist statistics
The Shapiro–Wilk test is a test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk. The Shapiro–Wilk test tests the null
Shapiro–Wilk_test
Nonparametric measure of rank correlation
confidence interval with level α {\displaystyle \alpha } is based on a Wilks' theorem given in the latter paper, and is given by { θ : { ∑ i = 1 n ( Z i −
Spearman's rank correlation coefficient
Spearman's_rank_correlation_coefficient
Notion in statistics
it is the Fubini–Study metric. It is the key part of the proof of Wilks' theorem, which allows confidence region estimates for maximum likelihood estimation
Fisher_information
Estimator for quality of a statistical model
criterion Maximum likelihood estimation Principle of maximum entropy Wilks' theorem Stoica, P.; Selen, Y. (2004), "Model-order selection: a review of information
Akaike_information_criterion
Statistical model tool
slightly different formulation suited to the use of log-likelihoods (see Wilks' theorem), the test statistic is twice the difference in log-likelihoods and
Relative_likelihood
Measure of goodness of fit for a statistical model
difference between the deviances for the two models follows, based on Wilks' theorem, an approximate chi-squared distribution with k-degrees of freedom.
Deviance_(statistics)
Theorem about the power of the likelihood ratio test
been Buster Keaton. Error exponents in hypothesis testing F-test Lemma Wilks' theorem Neyman, J.; Pearson, E. S. (1933-02-16). "IX. On the problem of the
Neyman–Pearson_lemma
Specialized form of regression analysis, in statistics
another, including unit weights, a result sometimes referred to as Wilks' theorem (Ree, Carretta, & Earles, 1998). Robyn Dawes (1979) examined decision
Robust_regression
Welch–Satterthwaite equation Well-behaved statistic Wick product Wilks' lambda distribution Wilks' theorem Winsorized mean Whipple's index White test White noise
List_of_statistics_articles
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
Family of functions to transform data
the Box–Cox transformation can be asymptotically constructed using Wilks's theorem on the profile likelihood function to find all the possible values
Power_transform
Equation for radii of tangent circles
1038/139062a0 Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R. (2002), "Beyond the Descartes circle theorem", The American Mathematical Monthly, 109 (4):
Descartes's_theorem
Statistical method for multiple testing
). For likelihood ratio tests with exactly two degrees of freedom, Wilks' theorem implies that p i = 1 / R i {\textstyle p_{i}=1/R_{i}} , where R i {\textstyle
Harmonic_mean_p-value
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
discussed in 1938 by Samuel Stanley Wilks, a leading statistician who had a special interest in multivariate analysis. Wilks described how unit weights could
Unit-weighted_regression
Probability distribution used in multivariate hypothesis testing
In statistics, Wilks' lambda distribution (named for Samuel S. Wilks), is a probability distribution used in multivariate hypothesis testing, especially
Wilks's_lambda_distribution
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
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
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
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
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
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
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)
American mathematician (1915–2000)
worked at the Fire Control Research Office and collaborated with Samuel Wilks and William Cochran. After the war, he returned to Princeton, dividing his
John_Tukey
Probability distribution
distributions are not known. Their importance is partly due to the central limit theorem. It states that the average of many statistically independent samples (observations)
Normal_distribution
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
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
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
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
Partial differential equation
Böhm and Wilking's method, they were able to derive a new Ricci flow convergence theorem (Brendle & Schoen 2009). Their convergence theorem included as
Ricci_flow
Indian-American mathematician (1920–2023)
Mahalanobis Centenary Gold Medal (1993?) of the Indian Science Congress Wilks Memorial Award (1989) of the American Statistical Association Padma Bhushan
C._R._Rao
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)
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
Unbiased statistical estimator minimizing variance
estimator δ ~ . {\displaystyle {\tilde {\delta }}.} Using the Rao–Blackwell theorem one can prove that determining the MVUE is simply a matter of finding a
Minimum-variance unbiased estimator
Minimum-variance_unbiased_estimator
Model for generating observable data in probability and statistics
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Generative_model
Probability distribution and special case of gamma distribution
of the test statistic approaches the normal distribution (central limit theorem). Because the test statistic (such as t) is asymptotically normally distributed
Chi-squared_distribution
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
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
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
Unit of information
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Data
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
Procedure for comparing multivariate sample means
Wilks' Λ Wilks = ∏ 1 , … , p ( 1 / ( 1 + λ p ) ) = det ( I + A ) − 1 = det ( S res ) / det ( S res + S model ) {\displaystyle \Lambda _{\text{Wilks}}=\prod
Multivariate analysis of variance
Multivariate_analysis_of_variance
Meaning represented by natural language
knowledge representation and automated reasoning (in particular, automated theorem proving). Since 1999 there has been an ACL special interest group on computational
Computational_semantics
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
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
Statistical hypothesis test
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Chi-squared_test
Measure of statistical dispersion
Bertil., Westergren (1988). Beta [beta] mathematics handbook : concepts, theorems, methods, algorithms, formulas, graphs, tables. Studentlitteratur. p. 348
Interquartile_range
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
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
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
Intelligence in machines
Alfred University Library. Retrieved 14 July 2026. Masterman, Margaret. Wilks, Yorick (ed.). "Language, Cohesion and Form" (PDF). Cambridge University
Artificial_intelligence
American mathematician (born 1950)
Böhm and Burkhard Wilking, thereby obtaining a new convergence theorem for Ricci flow. A special case of their convergence theorem has the differentiable
Richard_Schoen
Statistical hypothesis test
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
F-test
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
Value that appears most often in a set of data
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Mode_(statistics)
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
Type of chart
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Bar_chart
Graphical representation of the distribution of numerical data
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Histogram
American mathematician
received his Ph.D. at Princeton University in 1947 under advisor Samuel Wilks. From 1947 until 1966 he worked for the RAND Corporation, heading their
Ted_Harris_(mathematician)
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
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
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 matching technique
balancing score serves as a sufficient statistic for Z. Furthermore, the above theorems indicate that the propensity score is a minimal sufficient statistic if
Propensity_score_matching
Measure of the asymmetry of random variables
JSTOR 2684367. Johnson, NL, Kotz, S & Balakrishnan, N (1994) p. 3 and p. 40 Wilks DS (1995) Statistical Methods in the Atmospheric Sciences, p 27. Academic
Skewness
Probabilistic problem-solving algorithm
will be samples from the desired (target) distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures
Monte_Carlo_method
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)
Complete set of items that share at least one property in common
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Statistical_population
Statistical measure of association
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Cramér's_V
Generalization of the one-dimensional normal distribution to higher dimensions
distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least
Multivariate normal distribution
Multivariate_normal_distribution
Mathematical relation assigning a probability event to a cost
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Loss_function
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
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
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
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
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
Measure of variation in statistics
is at least as much as given in the following table. The central limit theorem states that the distribution of an average of many independent, identically
Standard_deviation
Statistical measure of variability
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Median_absolute_deviation
Statistical considerations on how many observations to make
this phenomenon, including the law of large numbers and the central limit theorem. In some situations, the increase in precision for larger sample sizes
Sample_size_determination
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
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
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
Comparison of two distributions
order statistic of a standard normal distribution. More generally, Shapiro–Wilk test uses the expected values of the order statistics of the given distribution;
Q–Q_plot
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
Class of statistical models
the use of the least-squares estimator is justified by the Gauss–Markov theorem, which does not assume that the distribution is normal. From the perspective
Generalized_linear_model
Concepts from statistical hypothesis testing
(mathematics) – Theorem for proving more complex theorems Jerzy Neyman – Polish American mathematician (1894–1981) Neyman–Pearson lemma – Theorem about the
Type_I_and_type_II_errors
Linear regression model with a single explanatory variable
normally distributed. The latter case is justified by the central limit theorem. Under the first assumption above, that of the normality of the error terms
Simple_linear_regression
Concept in inferential statistics
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Statistical_significance
Distinction between nominal, ordinal, interval and ratio variables
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Level_of_measurement
Numerical measure of a statistical relationship between variables
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Correlation_coefficient
Type of statistical measure over subsets of a dataset
{\displaystyle \lim _{\varepsilon \to 0}M_{f,\ \varepsilon }=f} by fundamental theorem of calculus and L'Hôpital's rule. Continuous moving average sine and polynom
Moving_average
Statistical measure of how far values spread from their average
standard error of the sample mean, which is used in the central limit theorem. To prove the initial statement, it suffices to show that Var ( X + Y
Variance
Relationship between items in a set
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Ranking
Criterion for model selection
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle
Bayesian information criterion
Bayesian_information_criterion
Fourth standardized moment in statistics
range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary
Kurtosis
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WILKS THEOREM
WILKS THEOREM
WILKS THEOREM
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WILKS THEOREM
WILKS THEOREM
WILKS THEOREM
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