Empirical Processes with Applications to Statistics

Shorack, Galen R.; Wellner, Jon A.

doi:10.1137/1.9780898719017

Cited by 1,177 publications

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“…To bound the right side of (4.16), we follow the symmetrization argument used in the proof of inequality 3 on p.820 of [36] (which in turn follows Lemma 1.1 of [27]). Let the sequence {η * i (τ n,i )} be an independent copy of {η i (τ n,i )} conditional on {τ n,i } and let ξ i be i.i.d.…”

Section: Step 3: Proof Of C-tightness Of {Xmentioning

confidence: 99%

“…We omit the proof because it is similar to that of Lemma (A.14.15) in [36], but we do give the proof in the appendix of the longer online version. …”

Section: Step 3: Proof Of C-tightness Of {Xmentioning

confidence: 99%

“…Proof of Lemma 4.2. We mimic the proof of (A.14.15) in [36]. First for a deterministic g ∈ D D , we have, for…”

Section: Appendix B: Useful Inequalitiesmentioning

confidence: 99%

“…As in [30], and in Louchard [26] and Krichagina and Puhalskii [20] before, we work with the empirical process of the service times. As can be seen from §2.2 of [41], §14 of [1] and especially Shorack and Wellner [36], empirical processes and associated statistical tests have been a major focus of FCLT's, ever since [4], so that there are many useful tools for queueing theory.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, to address the technical challenge of non-identically distributed remaining service times, we draw on Chapter 25 of [36], which in turn uses a symmetrization argument from [27], which can be traced back to [39]. Substantial new arguments are required as can be seen from the tightness proof in §4.2.3.…”

Section: Introductionmentioning

confidence: 99%

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Heavy-traffic Limit for the Initial Content Process

2017

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To understand the performance of a queueing system, it can be useful to focus on the evolution of the content that is initially in service at some time. That necessarily will be the case in service systems that provide service during normal working hours each day, with the system shutting down at some time, except that all customers already in service at the termination time are allowed to complete their service. Also, for infinite-server queues, it is often fruitful to partition the content into the initial content and the new input, because these can be analyzed separately. With i.i.d service times having a nonexponential distribution, the state of the initial content can be described by specifying the elapsed service times of the remaining initial customers. That initial content process is then a Markov process. This paper establishes a many-server heavy-traffic (MSHT) functional central limit theorem (FCLT) for the initial content process in the space D D , assuming a FCLT for the initial age process, with the number of customers initially in service growing in the limit. The proof applies a symmetrization lemma from the literature on empirical processes to address a technical challenge: For each time, including time 0, the conditional remaining service times, given the ages, are mutually independent but in general not identically distributed. (FCLT's) for the standard G/G/s queueing model, with unlimited waiting space and service in order of arrival, expose the impact of the stochastic variability in the arrival and service processes on the transient and steady-state performance. This is important because the general G/G/s model is far less tractable than its Markovian M/M/s counterpart, even for the special case in which the interarrival times and service times come from independent sequences of i.i.d. random variables. From [14,41], we know that conventional heavy-traffic theory tells a simple story: With conventional heavy-traffic, where the arrival rate increases to the maximum possible service rate with a fixed number of servers, the arrival and service processes contribute via Introduction. Heavy-traffic (HT) functional central limit theorems

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Section: Step 3: Proof Of C-tightness Of {Xmentioning

confidence: 99%

“…We omit the proof because it is similar to that of Lemma (A.14.15) in [36], but we do give the proof in the appendix of the longer online version. …”

Section: Step 3: Proof Of C-tightness Of {Xmentioning

confidence: 99%

“…Proof of Lemma 4.2. We mimic the proof of (A.14.15) in [36]. First for a deterministic g ∈ D D , we have, for…”

Section: Appendix B: Useful Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Heavy-traffic Limit for the Initial Content Process

2017

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Bootstrap Hypothesis Testing

MacKinnon

2009

Handbook of Computational Econometrics

121

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This paper surveys bootstrap and Monte Carlo methods for testing hypotheses in econometrics. Several different ways of computing bootstrap P values are discussed, including the double bootstrap and the fast double bootstrap. It is emphasized that there are many different procedures for generating bootstrap samples for regression models and other types of model. As an illustration, a simulation experiment examines the performance of several methods of bootstrapping the supF test for structural change with an unknown break point.

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