Edgeworth Expansions for Linear Combinations of Order Statistics with Smooth Weight Functions

Helmers, Roelof

doi:10.1214/aos/1176345207

Cited by 45 publications

(48 citation statements)

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“…Stigler provides a detailed treatment of the early history of L-estimators [22]. Broad summaries of the L-estimator literature can be found in [20], chapter 1 of [11] and chapter 22 of [24].…”

Section: Asymptotic Properties Ofμ I:nmentioning

confidence: 99%

On optimal system designs in reliability‐economics frameworks

Dugas

Samaniego

2007

Naval Research Logistics

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Reliability Economics is a field that can be defined as the collection of all problems in which there is tension between the performance of systems of interest and their cost. Given such a problem, the aim is to resolve the tension through an optimization process that identifies the system which maximizes some appropriate criterion function (e.g. expected lifetime per unit cost). In this paper, we focus on coherent systems of n independent and identically distributed (iid) components and mixtures thereof, and characterize both a system's performance and cost as functions of the system's signature vector (Samaniego, IEEE Trans Reliabil (1985) 69-72). For a given family of criterion functions, a variety of optimality results are obtained for systems of arbitrary order n. Approximations are developed and justified when the underlying component distribution is unknown. Assuming the availability of an auxiliary sample of N component failure times, the asymptotic theory of L-estimators is adapted for the purpose of establishing the consistency and asymptotic normality of the proposed estimators of the expected ordered failure times of the n components of the systems under study. These results lead to the identification of -optimal systems relative to the chosen criterion function.

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Section: Asymptotic Properties Ofμ I:nmentioning

confidence: 99%

On optimal system designs in reliability‐economics frameworks

Dugas

Samaniego

2007

Naval Research Logistics

View full text Add to dashboard Cite

show abstract

“…Edgeworth expansions. In his 1980 work [11], Helmers gave Edgeworth expansions for distributions of normalized L-statistics expressed by (2) and (3), which produce the error of order o(n −1 ). He achieved his results by analytic methods, under conditions including the existence of J (s) on (0, 1) and E|X 1 | 4 < ∞.…”

mentioning

confidence: 99%

“…In the proof we use probabilistic methods and apply an already known result for U -statistics. In comparison with the work of Helmers [11] we weaken the conditions concerning the smoothness of J(s) but we put some additional requirements on the distribution of X 1 .…”

mentioning

confidence: 99%

“…To answer this question we compare the Edgeworth series for the mean h n and for the L-statistic I n . The Edgeworth series for the statistics I n and E n (see (3) and (2)) can be found in [11]. They were obtained under assumptions on the smoothness of J(s) (the existence of J (s)) and on the existence of E|X 1 | 4 .…”

mentioning

confidence: 99%

“…(see [11], p. 1363). Since Eh(X 1 ) = 0 and Var h(X 1 ) = σ 2 , the Edgeworth series for the mean h n is…”

mentioning

confidence: 99%

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