Expansions about the Gamma for the Distribution and Quantiles of a Standard Estimate

Withers, Christopher S.; Nadarajah, Saralees

doi:10.1007/s11009-013-9328-9

Cited by 3 publications

(7 citation statements)

References 22 publications

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“…It also allowed for expansions about asymptotically normal random variables. The advantage of this approach in greatly reducing the number of terms in each P r (x) was illustrated in [7].…”

Section: Discussionmentioning

confidence: 99%

“…Therefore, to derive the 5th-order Edgeworth expansion for the distribution of n 1/2 (t( ŵ) − t(w)) for ŵ a standard estimate, we simply substitute the coefficients in ( 6) and (7) in the expression for P r (x), r ≤ 4, with those corresponding to t( ŵ) as provided in Sections 3-5. Equation ( 9) of [3] provides P r (x) for the more general case where P 0 (x) is the distribution function of Y in R p which depends on n but is asymptotic to Φ V (x) and has a Type B expansion.…”

Section: Introductionmentioning

confidence: 99%

“…Equation ( 9) of [3] provides P r (x) for the more general case where P 0 (x) is the distribution function of Y in R p which depends on n but is asymptotic to Φ V (x) and has a Type B expansion. One can choose P 0 (x) so that the number of terms in each P r (x) greatly reduces: see Withers and Nadarajah (2012d) [7,8]. When ŵ is lattice, further terms need to be added: see for example Chapter 5 of [9], [10], and for the density of Y nw , p211 of [11], Section 5 of [12], and Section 6 of [13].…”

Section: Introductionmentioning

confidence: 99%

“…Cumulant coefficients are also needed for bias reduction, Bayesian inference, confidence regions and power. See [7,8,[14][15][16][17][18]. for examples.…”

Section: Introductionmentioning

confidence: 99%

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5th-Order Multivariate Edgeworth Expansions for Parametric Estimates

Withers

2024

Mathematics

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The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the Edgeworth expansion for the distribution of the standardised estimate. This is an expansion in n−1/2 about the normal distribution, where n is typically the sample size. The first few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for any standard estimate, hugely expanding its application. We define an estimate w^ of an unknown vector w in Rp, as a standard estimate, if Ew^→w as n→∞, and for r≥1 the rth-order cumulants of w^ have magnitude n1−r and can be expanded in n−1. Here we present a significant extension. We give the expansion of the distribution of any smooth function of w^, say t(w^) in Rq, giving its distribution to n−5/2. We do this by showing that t(w^), is a standard estimate of t(w). This provides far more accurate approximations for the distribution of t(w^) than its asymptotic normality.

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“…It also allowed for expansions about asymptotically normal random variables. The advantage of this approach in greatly reducing the number of terms in each P r (x) was illustrated in [7].…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Cumulant coefficients are also needed for bias reduction, Bayesian inference, confidence regions and power. See [7,8,[14][15][16][17][18]. for examples.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

5th-Order Multivariate Edgeworth Expansions for Parametric Estimates

Withers

2024

Mathematics

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“…We can also assume the PDF of the final residual settlement S rf follows gamma (Γ) distribution and the shape parameter is greater than 1, namely, ∼ Γ( , ). According to [37], the shape and scale parameters and can be calculated and we have = 0.805 and the scale parameter = 10.05. Hence, the assumption is not satisfied since is smaller than 1.…”

Section: Analysis Of Characteristics Of Final Residual Settlementmentioning

confidence: 99%

A Novel Incipient Evaluation Method for Postconstruction Settlement by Using a Statistic Approach with Time‐Extended Loading Test

Hong

Shan

Liu

et al. 2019

Mathematical Problems in Engineering

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This paper proposes an incipient assessment method for postconstruction settlement of highway subgrade by using statistical analysis approach with an improved static loading test method, time-extended loading test. The time-extended loading residual settlement Sr(t) and settlement rate Vs(t) are calculated from the field test data. The test procedure and the corresponding experimental validations are presented. Based on the field test data, a probabilistic model is built to bridge the final residual settlement and the actual postconstruction settlements. The proposed time-extended loading test based assessment system has been validated through three operating and well-documented road sections. According to the validation results, the construction quality can be accurately evaluated at an early stage and the corresponding remedial measures can be applied timely.

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Quantile mechanics: Issues arising from critical review

Okagbue¹

2019

Int. j. adv. appl. sci.

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Approximations are the alternative way of obtaining the Quantile function when the inversion method cannot be applied to distributions whose cumulative distribution functions do not have close form expressions. Approximations come in form of functional approximation, numerical algorithm, closed form expressed in terms of others and series expansions. Several quantile approximations are available which have been proven to be precise, but some issues like the presence of shape parameters, inapplicability of existing methods to complex distributions and low computational speed and accuracy place undue limitations to their effective use. Quantile mechanics (QM) is a series expansion method that addressed these issues as evidenced in the paper. Quantile mechanics is a generalization of the use of ordinary differential equations (ODE) in quantile approximation. The paper is a review that critically examined with evidences; the formulation, applications and advantages of QM over other surveyed methods. Some issues bothering on the use of QM were also discussed. The review concluded with areas of further studies which are open for scientific investigation and exploration.

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Expansions about the Gamma for the Distribution and Quantiles of a Standard Estimate

Cited by 3 publications

References 22 publications

5th-Order Multivariate Edgeworth Expansions for Parametric Estimates

5th-Order Multivariate Edgeworth Expansions for Parametric Estimates

A Novel Incipient Evaluation Method for Postconstruction Settlement by Using a Statistic Approach with Time‐Extended Loading Test

Quantile mechanics: Issues arising from critical review

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