Edgeworth expansion
E174595
Edgeworth expansion is an asymptotic series that refines the central limit theorem by providing higher-order approximations to the distribution of normalized sums of random variables.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Edgeworth expansion canonical | 1 |
| Gram–Charlier series | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1535800 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Edgeworth expansion Context triple: [Berry–Esseen theorem, relatedTo, Edgeworth expansion]
-
A.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
B.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
-
C.
Frisch–Waugh–Lovell theorem
The Frisch–Waugh–Lovell theorem is a fundamental result in econometrics that shows how the coefficients of a multiple linear regression can be obtained by first partialling out (regressing out) other explanatory variables.
-
D.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
E.
Laplace law of error
The Laplace law of error is a probability distribution characterized by a sharp peak at the mean and heavier tails than the normal distribution, historically used to model the magnitude of observational errors.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Edgeworth expansion Target entity description: Edgeworth expansion is an asymptotic series that refines the central limit theorem by providing higher-order approximations to the distribution of normalized sums of random variables.
-
A.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
B.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
-
C.
Frisch–Waugh–Lovell theorem
The Frisch–Waugh–Lovell theorem is a fundamental result in econometrics that shows how the coefficients of a multiple linear regression can be obtained by first partialling out (regressing out) other explanatory variables.
-
D.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
E.
Laplace law of error
The Laplace law of error is a probability distribution characterized by a sharp peak at the mean and heavier tails than the normal distribution, historically used to model the magnitude of observational errors.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
asymptotic expansion
ⓘ
probability theory concept ⓘ statistical approximation method ⓘ |
| aimsTo | approximate tail probabilities more accurately than the normal approximation ⓘ |
| appliesTo | normalized sums of random variables ⓘ |
| approximates |
distribution function of standardized sums
ⓘ
probability density function of standardized sums ⓘ |
| assumes | independent and identically distributed summands in its classical form ⓘ |
| canBeExtendedTo | certain dependent data settings ⓘ |
| canBeWrittenAs | series involving standardized cumulants ⓘ |
| converges | asymptotically rather than absolutely in general ⓘ |
| field |
mathematical statistics
ⓘ
probability theory ⓘ |
| generalizes | normal approximation in the central limit theorem ⓘ |
| hasOrder |
first-order correction beyond the central limit theorem
ⓘ
second-order correction beyond the central limit theorem ⓘ |
| historicalDevelopment | originated in early 20th century work of Edgeworth and others ⓘ |
| improves | accuracy of normal approximation for finite samples ⓘ |
| includes |
kurtosis corrections
ⓘ
skewness corrections ⓘ |
| is |
an asymptotic series in powers of n^{-1/2}
ⓘ
typically truncated after a finite number of terms in applications ⓘ |
| isDiscussedIn | advanced textbooks on asymptotic statistics ⓘ |
| isOftenDerivedUsing |
characteristic functions
ⓘ
cumulant generating functions ⓘ |
| isRelatedTo |
Berry–Esseen theorem
ⓘ
Edgeworth expansion self-linksurface differs ⓘ
surface form:
Gram–Charlier series
saddlepoint approximation ⓘ |
| isUsedFor |
refined approximations to confidence interval coverage
ⓘ
refined approximations to distribution quantiles ⓘ refined approximations to p-values ⓘ |
| isUsedIn |
approximating sampling distributions
ⓘ
asymptotic statistics ⓘ higher-order asymptotics ⓘ statistical inference ⓘ |
| isValidAs | asymptotic approximation as sample size tends to infinity ⓘ |
| may | produce negative probabilities when truncated ⓘ |
| namedAfter | Francis Ysidro Edgeworth ⓘ |
| provides | higher-order approximations to probability distributions ⓘ |
| refines | central limit theorem ⓘ |
| requires |
existence of sufficiently high-order moments
ⓘ
regularity conditions on the underlying distribution ⓘ |
| uses |
Hermite functions
ⓘ
surface form:
Hermite polynomials
cumulants of random variables ⓘ moments of random variables ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Edgeworth expansion Description of subject: Edgeworth expansion is an asymptotic series that refines the central limit theorem by providing higher-order approximations to the distribution of normalized sums of random variables.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gram–Charlier series