Statistic
Builds on Random variableIndependent and identically distributed random variables
Definition
Let be random variables with values in a measurable space . For example, they can be a random sample from a law with an unknown parameter . A statistic is a random variable of the form , where is a measurable function that does not depend on .
Remarks
Examples are the sample mean , the sample variance, the sample maximum and the order statistics. A statistic is a random variable, so it has a law: the sampling distribution of the statistic. A statistic that one uses to estimate a parameter is an estimator. A statistic that one uses to decide a hypothesis test is a test statistic.
Reading path
To understand this page from the ground up, read these 48 pages in order. The foundations come first.
- Construction of the reals from Cauchy sequences concept
- Element definition
- Integers object
- Ordered field definition
- Proposition concept
- Random experiment concept
- Rational numbers object
- Real numbers object
- Set definition
- Ring of integers statement
- Disjunction definition
- Implication definition
- Predicate definition
- Ordered pair definition
- Union definition
- Sample space definition
- The rationals form a field statement
- Existential quantifier definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- The rationals form an ordered field statement
- Uniqueness quantifier definition
- Binary relation definition
- Universal set definition
- Order on the reals definition
- Power set definition
- Complement definition
- Function definition
- Interval definition
- Preimage of a set definition
- Sigma-algebra definition
- Event definition
- Borel sigma-algebra definition
- Probability axioms concept
- Probability measure definition
- Probability distribution definition
- Probability space definition
- Conditional probability definition
- Random variable definition
- Independence of two events definition
- Law of a random variable definition
- Mutual independence of events definition
- Identically distributed random variables definition
- Independent sigma-algebras definition
- Independence of random variables definition
- Independent and identically distributed random variables definition
- Statistic definition
Where this leads
97 pages build on this one.
- Bias of an estimator
- Statistical hypothesis test
- Method of moments
- Neyman–Pearson lemma
- P-value
- Chi-squared test
- Maximum likelihood estimation
- Student's t-test
- Sample variance
- Confidence interval
- Student's t-distribution
- Sample mean
- Type I and type II errors
- Chi-squared distribution
- Estimator
- Mean squared error
- Consistent estimator
- Efficiency of an estimator
- Log-rank test
- Poisson regression
- Sequential analysis
- Hodges' superefficient estimator
- Mathematical statistics
- F-test
- Wald–Wolfowitz optimality of the sequential probability ratio test
- Sequential probability ratio test
- Consistency of the maximum likelihood estimator
- Basu's theorem
- Order statistic
- Lindley's paradox
- M-estimator
- Score test
- Multivariate analysis of variance
- Asymptotic normality of the maximum likelihood estimator
- Mann–Whitney U test
- F-distribution
- Monotone likelihood ratio
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Minimal sufficient statistic
- Whittle likelihood
- Logistic regression
- Cramér–von Mises criterion
- Robust statistics
- Bias–variance decomposition
- Breakdown point
- Noncentral chi-squared distribution
- Hájek–Le Cam convolution theorem
- Behrens–Fisher problem
- Hodges–Lehmann estimator
- Kruskal–Wallis test
- Sampling distribution
- Wishart distribution
- Statistical inference
- Minimum-variance unbiased estimator
- Sign test
- Empirical likelihood
- Bayesian information criterion
- Cochran's theorem
- Jackknife
- Akaike information criterion
- Independence of sample mean and sample variance for normal samples
- Estimating equations
- Anderson–Darling test
- Family-wise error rate
- Dickey–Fuller test
- Generalized method of moments
- Uniformly most powerful test
- Empirical distribution function
- Bootstrap
- Ancillary statistic
- Complete statistic
- Horvitz–Thompson estimator
- U-statistic
- L-estimator
- Power of a test
- Sufficient statistic
- Likelihood-ratio test
- Karlin–Rubin theorem
- Kolmogorov–Smirnov test
- Expectation–maximization algorithm
- Hotelling's T-squared distribution
- Cross-validation
- Wald test
- Generalized linear model
- Box–Jenkins method
- Binomial proportion confidence interval
- False discovery rate and the Benjamini–Hochberg procedure
- Neyman construction
- Wilks' theorem
- Permutation test
- Fisher's exact test
- Bonferroni correction
- Frequentist inference
- Fisher–Neyman factorization theorem
- Observed information
- Point estimation
Further reading
- Web https://en.wikipedia.org/wiki/Statistic, lead section
About this page
| Tag | 036756NFD6 |
|---|---|
| URI | m2://036756NFD6 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 62A01 |
| Defines | Statistic |
| Uses | Random variable, Independent and identically distributed random variables |
Markdown · JSON · Edit on GitHub
What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Used by
- Statistical hypothesis test
- P-value
- Sample variance
- Confidence interval
- Sample mean
- Estimator
- Mathematical statistics
- F-test
- Order statistic
- Monotone likelihood ratio
- Sample mean and sample covariance
- Sampling distribution
- Statistical inference
- Jackknife
- Empirical distribution function
- Bootstrap
- Ancillary statistic
- Complete statistic
- U-statistic
- Sufficient statistic
- Karlin–Rubin theorem
- Permutation test
- Point estimation