Tabulating weighted sup-norm functionals used in change-point analysis

Orasch, Markus; Pouliot, William

doi:10.1080/0094965031000148703

Cited by 12 publications

(14 citation statements)

References 3 publications

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“…This gives us an asymptotically correct 100(1 − α)% confidence band [L n (t), U n (t)] for F (t) on the interval t ∈ [X (1) , X (n) ), where L n (t) = max 0, F n (t) − c α √ n q(F n (t)) , U n (t) = min 1, F n (t) + c α √ n q(F n (t)) , and, given α ∈ (0, 1), the value of c α is found from Table III in [26]. For instance, c 0.05 = 4.57.…”

Section: Confidence Bandsmentioning

confidence: 99%

“…The distribution of the random variable sup 0<u<1 |B(u)|/q(u) has been tabulated in B. Eastwood and V. Eastwood [15] and, using a somewhat different approach, in Orasch and Pouliot [26]. In this section, the tabulation of the distribution of sup 0<u<1 B(u)/q(u) follows the approach of [26]. Namely, we shall use the following algorithm.…”

Section: Tabulation Of Cumulative Distribution Functionsmentioning

confidence: 99%

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Goodness-of-fit tests based on sup-functionals of weighted empirical processes

Stepanova¹,

Pavlenko²

2018

Teoriya Veroyatnostei i ee Primeneniya

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A large class of goodness-of-fit test statistics based on sup-functionals of weighted empirical processes is proposed and studied. The weight functions employed are Erdős-Feller-Kolmogorov-Petrovski upper-class functions of a Brownian bridge. Based on the result of M. Csörgő, S. Csörgő, Horváth, and Mason obtained for this type of test statistics, we provide the asymptotic null distribution theory for the class of tests in hand, and present an algorithm for tabulating the limit distribution functions under the null hypothesis. A new family of nonparametric confidence bands is constructed for the true distribution function and it is found to perform very well. The results obtained, together with a new result on the convergence in distribution of the higher criticism statistic, introduced by Donoho and Jin, demonstrate the advantage of our approach over a common approach that utilizes a family of regularly varying weight functions. Furthermore, we show that, in various subtle problems of detecting sparse heterogeneous mixtures, the proposed test statistics achieve the detection boundary found by Ingster and, when distinguishing between the null and alternative hypotheses, perform optimally adaptively to unknown sparsity and size of the non-null effects.

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Section: Confidence Bandsmentioning

confidence: 99%

Section: Tabulation Of Cumulative Distribution Functionsmentioning

confidence: 99%

Goodness-of-fit tests based on sup-functionals of weighted empirical processes

Stepanova¹,

Pavlenko²

2018

Teoriya Veroyatnostei i ee Primeneniya

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“…The weight function q(·) is to emphasize changes that may have recurred near 0 and n. We note in passing that the result in (1.7) cannot be deduced via first proving a "corresponding" weak invariance principle on D[0, 1] (cf. The applicability of (1.7) is much enhanced by Orasch and Pouliot (2004), tabulating functionals in weighted sup-norm.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We only prove (2.6) and (2.8), for the proof of (2.7) is similar to that of (2.6). Since η n is a regularly varying function with index 1/2, we have that for sufficiently large n, η n/(log log n) 16 (log log n) 3 ≤ η n/(log log n) 9 .…”

Section: Proofs Of Theorems 11 and 12mentioning

confidence: 99%

Change in the mean in the domain of attraction of the normal law via Darling–Erdős theorems

Csörgő¹,

Hu²

2014

Braz. J. Probab. Stat.

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This paper studies the problem of testing the null assumption of no-change in the mean of chronologically ordered independent observations on a random variable X versus the at most one change in the mean alternative hypothesis. The approach taken is via a Darling-Erdős type self-normalized maximal deviation between sample means before and sample means after possible times of a change in the expected values of the observations of a random sample. Asymptotically, the thus formulated maximal deviations are shown to have a standard Gumbel distribution under the null assumption of no change in the mean. A first such result is proved under the condition that EX 2 log log(|X| + 1) < ∞, while in the case of a second one, X is assumed to be in a specific class of the domain of attraction of the normal law, possibly with infinite variance.Key Words: Change in the mean, domain of attraction of the normal law, Darling-Erdős theorems, Gumbel distribution, weighted metrics, Brownian bridge.

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“…The convergence result in Fact 1 continues to hold when q(F (t)) is replaced by q(F n (t)) (see Remark 3.1 in Stepanova and Pavlenko [27]). The distribution of the limiting random variable sup 0<u<1 |B(u)|/q(u) was tabulated in Eastwood and Eastwood [12] and later, using somewhat different method, in Orasch and Pouliot [25].…”

Section: Preliminariesmentioning

confidence: 99%

Estimation of the Amount of Sparsity in Normal Mixture Models

Wang¹

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In this thesis, we are interested in estimating the amount of sparsity in sparse normal mixture models. The estimation problem in hand emerges naturally in the context of variable selection in high-dimensional settings. Handling this problem, we have modified the wellknown procedure of Cai et al. (2007) on estimating the proportion of nonzero means in normal mixture models in such a way that the new procedure is more efficient in theory, less complicated in construction, and less asymptotic in applications. The analytical findings obtained in the thesis are supported via simulations. The simulation study testifies that the new procedure not only displays a better convergence rate but also requires smaller sample size in order to work as designed. Theorem 7 of Chapter 3 that establishes the upper bound on the risk of the newly proposed estimator is the main result of the thesis. This result shows the superiority of our estimator over the estimator of Cai et al. (2007), and it is new.

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Tabulating weighted sup-norm functionals used in change-point analysis

Cited by 12 publications

References 3 publications

Goodness-of-fit tests based on sup-functionals of weighted empirical processes

Goodness-of-fit tests based on sup-functionals of weighted empirical processes

Change in the mean in the domain of attraction of the normal law via Darling–Erdős theorems

Estimation of the Amount of Sparsity in Normal Mixture Models

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