Field theory and polynomials Definition
Ordered field
Definition
An ordered field is a field with a total order on . The order must have these two properties for all , and in :
- if , then ;
- if and , then .
Remarks
In an ordered field, each square is nonnegative: for each . In particular and . An equivalent definition uses a positive cone . This is a subset of that is closed under addition and multiplication. Also, must be the disjoint union of , and . The field of complex numbers has no such order.
Where this leads
428 pages build on this one.
- Variance
- Prior probability
- Absolute value
- Chernoff bound
- Chapman–Kolmogorov equation
- Bias of an estimator
- The complex numbers cannot be ordered
- Identically distributed random variables
- Law of the unconscious statistician
- Density of the normal distribution
- Exponential distribution
- Covariance matrix
- Chebyshev's inequality
- Characteristic function of a sum of independent random variables
- Stationary distribution
- Bernoulli process
- Order on the reals
- Markov chain
- Statistic
- Law of total variance
- Variance of the normal distribution
- Triangle inequality
- Cauchy–Schwarz inequality for expectations
- Monotonicity of probability
- Every real number has a decimal expansion
- Correlation coefficient
- Variance under translation
- Density of the irrationals
- Almost sure convergence
- Dedekind cut
- Law of total expectation
- Probability density function
- Expectation of a product of independent random variables
- Likelihood function
- Markov's inequality
- Standard score
- Poisson point process
- Gamma distribution
- De Moivre–Laplace theorem
- Buffon's needle problem
- Kolmogorov's zero–one law
- Independent random variables are uncorrelated
- Convolution of probability distributions
- Normal distribution
- Mean of the Poisson distribution
- Statistical hypothesis test
- Independence means the joint law is the product of the marginals
- Median
- Geometric probability
- Intermediate value theorem for connected spaces
- Dirac measure
- Method of moments
- Mean of the normal distribution
- The rationals form an ordered field
- Independent sigma-algebras
- The reals are Archimedean
- Archimedean property
- Independent and identically distributed random variables
- Classical definition of probability
- Pairwise independent events that are not mutually independent
- Ceiling function
- Marginal distribution
- Moment-generating function of the normal distribution
- Quantile function
- Connected subsets of the real line are intervals
- Weak law of large numbers
- Multiplication rule
- Borel sigma-algebra
- Addition rule for two events
- Random walk
- Random variable with a density
- Uncorrelated random variables that are not independent
- Reverse triangle inequality
- Kurtosis
- Martingale
- Variance of a sum
- Mean of the geometric distribution
- Expected value
- Strong law of large numbers
- Covariance
- Random variable
- Base rate fallacy
- Law of total probability
- Independence of two random variables
- Neyman–Pearson lemma
- Convergence in probability
- Boole's inequality
- P-value
- Chi-squared test
- Convergence in probability implies convergence in distribution
- Central limit theorem
- 68–95–99.7 rule
- Transition matrix
- Conditional variance
- Conditional probability
- Lévy's continuity theorem
- Convergence of random variables
- Maximum likelihood estimation
- Multivariate normal distribution
- Uniqueness of the complete ordered field
- Mutual independence of events
- Absolute value is multiplicative
- Skewness
- Galton–Watson process
- Markov property
- Archimedean ordered field
- Student's t-test
- Discrete random variable
- Bernoulli distribution
- Convergence in distribution
- The cumulative distribution function determines the distribution
- Sample variance
- Bayesian inference
- Confidence interval
- Posterior probability
- Variance under scaling
- Discrete uniform distribution
- Coupon collector's problem
- Cumulative distribution function
- Characteristic function
- Pairwise independence
- Cauchy distribution
- Bayes' theorem
- Variance as the mean of the square minus the square of the mean
- Birthday problem
- Moment-generating function
- Student's t-distribution
- Characteristic function of the normal distribution
- Second Borel–Cantelli lemma
- Sum of independent normal random variables
- Independence of two events
- Linearity of expectation
- Monty Hall problem
- Moment
- Beta distribution
- Probability of the complement
- Expectation with respect to the binomial distribution
- Cumulant generating function
- Joint probability distribution
- Probability distribution
- Stochastic process
- Sample mean
- The reals form an ordered field
- Type I and type II errors
- Almost surely
- Characterization of the floor
- Generating function of a sum of independent random variables
- Chi-squared distribution
- Limit of the cumulative distribution function at infinity
- Hypergeometric distribution
- Moments are derivatives of the moment-generating function at zero
- Density of the rationals
- Almost sure convergence implies convergence in probability
- Bertrand paradox
- First Borel–Cantelli lemma
- The characteristic function determines the distribution
- Interval
- Law of a random variable
- Convergence in mean
- Estimator
- Variance of a sum of independent random variables
- Mean squared error
- Poisson distribution
- Probability measure
- Binomial distribution
- Standard deviation
- Law of total probability for an event and its complement
- Probability mass function
- Gambler's ruin
- Continuous uniform distribution
- There is a natural number n with 1/n below any positive epsilon
- Hoeffding's inequality
- Moment-generating function of a sum of independent random variables
- Continuity of probability measures
- Independence of random variables
- Geometric distribution
- Monotonicity of the cumulative distribution function
- Consistent estimator
- Poisson limit theorem
- Negative binomial distribution
- Tail sum formula for expectation
- Conditional expectation
- Absorbing Markov chain
- Sum of independent Poisson random variables
- Conditional probability distribution
- Floor function
- Probability-generating function
- Memorylessness
- Conditional independence
- Path-connected spaces are connected
- Central moment
- Law of large numbers
- Probability space
- Doob's upcrossing inequality
- Bernstein–von Mises theorem
- Cramér–Rao bound
- Efficiency of an estimator
- Log-rank test
- Functional data analysis
- Poisson regression
- Sequential analysis
- Ergodic decomposition
- Formally real field
- Koksma–Hlawka inequality
- Hodges' superefficient estimator
- Martingale representation theorem
- Absorbing set
- Mathematical statistics
- F-test
- Laplace transform of the Dirac delta
- Stationary process
- Orthogonal polynomials on the unit circle
- Cramér–Wold theorem
- Boolean model
- Stochastic partial differential equation
- Wald–Wolfowitz optimality of the sequential probability ratio test
- Doob's martingale inequality
- Seminorm
- Sequential probability ratio test
- Brownian bridge
- Consistency of the maximum likelihood estimator
- Wiener process
- Empirical Bayes method
- Karhunen–Loève theorem
- Brent's method
- Itô isometry
- Absolute continuity
- Bayes factor
- Pickands–Balkema–De Haan theorem
- Basu's theorem
- Variogram
- Robbins–Monro algorithm
- Contiguity of measures
- Order statistic
- Balanced set
- Dirichlet process
- Lindley's paradox
- Universally measurable set
- M-estimator
- Ornstein–Uhlenbeck process
- Kernel regression
- Bayes estimator
- Bisection method
- Adapted process
- Score test
- Kingman's subadditive ergodic theorem
- Multivariate analysis of variance
- McDiarmid's inequality
- Lévy's characterization of Brownian motion
- Asymptotic normality of the maximum likelihood estimator
- Mann–Whitney U test
- Kolmogorov automorphism
- Score
- F-distribution
- Edgeworth expansion
- Crofton formula
- Monotone likelihood ratio
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Metropolis–Hastings algorithm
- Subderivative
- Continuous-time Markov chain
- Darboux function
- Kriging
- Log-convexity of the gamma function
- Identifiability
- Baire function
- Minimal sufficient statistic
- Whittle likelihood
- Logistic regression
- Convergence to equilibrium of ergodic Markov chains
- Cramér–von Mises criterion
- Azuma's inequality
- Gaussian free field
- Parry measure
- Kernel density estimation
- Erdős–Kac theorem
- Detailed balance
- Coupling from the past
- Robust statistics
- Hölder's theorem on Archimedean groups
- Hidden Markov model
- Finite-dimensional distributions
- Continuous mapping theorem
- Bias–variance decomposition
- Lévy process
- Wald's equation
- Random ordinary differential equation
- Monte Carlo integration
- Time series
- Optional stopping theorem
- Measurable space
- Breakdown point
- Delta method
- Minimum description length
- Completeness of the real numbers
- Noncentral chi-squared distribution
- Hájek–Le Cam convolution theorem
- Logarithmically convex function
- Harmonic measure
- Behrens–Fisher problem
- Submartingale
- Hodges–Lehmann estimator
- Kruskal–Wallis test
- Stochastic difference equation
- Sampling distribution
- Information geometry
- Recurrence and transience of states
- Malliavin calculus
- Wishart distribution
- Birth–death process
- Karamata's inequality
- Bayes risk
- Statistical inference
- Minimum-variance unbiased estimator
- Sign test
- Slutsky's theorem
- Tanaka's formula
- Geometric Brownian motion
- Empirical likelihood
- Dvoretzky–Kiefer–Wolfowitz inequality
- Bayesian information criterion
- Berry–Esseen theorem
- Continuity of measure from below and above
- Legendre transformation
- Hunt process
- Cochran's theorem
- Stochastic approximation
- Jackknife
- Quantile regression
- Markov chain mixing time
- Birkhoff's ergodic theorem
- Akaike information criterion
- Ordered ring
- Hewitt–Savage zero–one law
- Jensen's inequality for integrals
- Statistical model
- Bramble–Hilbert lemma
- Reflection principle
- Credible interval
- Conformal modulus
- Independence of sample mean and sample variance for normal samples
- Self-similar process
- Gibbs sampling
- Estimating equations
- Real closure
- Supermartingale
- Anderson–Darling test
- Spearman's rank correlation coefficient
- Family-wise error rate
- Markov chain Monte Carlo
- Nested intervals theorem
- Dickey–Fuller test
- Generalized method of moments
- Doob's martingale convergence theorems
- Random integral equation
- Uniformly most powerful test
- Bernstein inequalities
- Kendall rank correlation coefficient
- Semimartingale
- Empirical distribution function
- Bootstrap
- Minimax approximation
- Influence function
- Oseledets theorem
- Itô calculus
- Ancillary statistic
- Stopped process
- Jeffreys prior
- Moments of the log-normal distribution
- Complete statistic
- Horvitz–Thompson estimator
- U-statistic
- Fractional Brownian motion
- Bayesian hierarchical model
- L-estimator
- Power of a test
- Parametric model
- Randomization
- Sufficient statistic
- Gaussian process
- Missing data mechanisms
- Likelihood-ratio test
- Karlin–Rubin theorem
- Conjugate prior
- Kolmogorov–Smirnov test
- Expectation–maximization algorithm
- Hotelling's T-squared distribution
- Burkholder–Davis–Gundy inequality
- Doob–Meyer decomposition theorem
- Cross-validation
- Barbier's theorem
- Wald test
- Extremal length
- Pivotal quantity
- Generalized linear model
- Random graph
- Tame Fréchet space
- Box–Jenkins method
- Kolmogorov continuity theorem
- Shrinkage estimator
- Pesin theory
- Empirical process
- Binomial proportion confidence interval
- Lévy's upward theorem
- False discovery rate and the Benjamini–Hochberg procedure
- Particle filter
- Neyman construction
- Wilks' theorem
- Differentiable curve
- Skorokhod's embedding theorem
- Blaschke–Lebesgue theorem
- Permutation test
- Fisher's exact test
- Artin–Schreier theorem on formally real fields
- Bonferroni correction
- Doob decomposition theorem
- Local asymptotic normality
- Frequentist inference
- Likelihood principle
- Chebyshev nodes
- Fisher–Neyman factorization theorem
- Girsanov theorem
- Quadratic variation
- Observed information
- Feller process
- Point estimation
- Local martingale
Further reading
- Web https://en.wikipedia.org/wiki/Ordered_field, Total order
About this page
| Tag | 010JCV9J4E |
|---|---|
| URI | m2://010JCV9J4E |
| Kind | definition |
| Writing | W1 stated |
| MSC | 12J15 |
| Defines | Ordered field |
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What this page needs for W2
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