Boundary Non-crossings of Brownian Pillow

Hashorva, Enkelejd

doi:10.1007/s10959-008-0191-5

Cited by 9 publications

(16 citation statements)

References 33 publications

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“…d) Consider the additive Brownian pillow the conditions for f and u accordingly. Note that compared to [12] we do not need to put restrictions on f .…”

Section: Resultsmentioning

confidence: 99%

“…Numerous applications concerned with the evaluation of boundary non-crossing probabilities relate to mathematical finance, risk theory, queueing theory, statistics, physics among many other fields. Also calculation of boundary noncrossing probabilities of random fields are considered in various contexts, see e.g., [19,10,12,21]. Unlike the previous papers, we consider in this contribution the general model consisting of three components that include a standard Brownian sheet and two independent Wiener processes.…”

Section: Introductionmentioning

confidence: 99%

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Boundary noncrossings of additive Wiener fields∗

Hashorva

Mishura

2014

Lith Math J

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Let W i = {W i (t), t ∈ R + }, i = 1, 2 be two Wiener processes and W 3 = {W 3 (t), t ∈ R 2 + } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary non-crossing probabilitywhere h, u : R 2 + → R + are two measurable functions. We show further that for large trend functions γf > 0 asymptotically when γ → ∞ we have that ln P γf is the same as ln P γf where f is the projection of f on some closed convex set of the reproducing kernel Hilbert Space of W . It turns out that our approach is applicable also for the additive Brownian pillow.

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“…d) Consider the additive Brownian pillow the conditions for f and u accordingly. Note that compared to [12] we do not need to put restrictions on f .…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Boundary noncrossings of additive Wiener fields∗

Hashorva

Mishura

2014

Lith Math J

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“…In this section other formulas for the upper and lower bounds of the power of KS and CvM tests involving thedimensional set-indexed Brownian sheet and pillow models are derived. Our results are obtained by generalizing the approach proposed in that studied in [18,19] who confined the investigation to one-dimensional Kolmogorov type boundary noncrossing probability involving the so-called univariate ordinary Brownian sheet and pillow.…”

Section: Alternative Approachesmentioning

confidence: 93%

“…As suggested in [14], p. 315, and [15], p. 423-424, studying the power function is of importance to be able to evaluate the performance of the test especially their rate of decay to . Therefore in this paper we investigate the upper and lower bounds for (6) by considering the result for the univariate Brownian sheet and Brownian pillow presented in Janssen [17] and Hashorva [18,19]. Upper and lower bound for the power function of goodness-of-fit test involving multiparameter Brownian process have been studied by Bass [20].…”

Section: Introductionmentioning

confidence: 99%

Accessing the Power of Tests Based on Set-Indexed Partial Sums of Multivariate Regression Residuals

Somayasa

2018

Journal of Applied Mathematics

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The intention of the present paper is to establish an approximation method to the limiting power functions of tests conducted based on Kolmogorov-Smirnov and Cramér-von Mises functionals of set-indexed partial sums of multivariate regression residuals. The limiting powers appear as vectorial boundary crossing probabilities. Their upper and lower bounds are derived by extending some existing results for shifted univariate Gaussian process documented in the literatures. The application of multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions is demonstrated. The rate of decay of the power function to a presigned value α is also studied. Our consideration is mainly for the trend plus signal model including multivariate set-indexed Brownian sheet and pillow. The simulation shows that the approach is useful for analyzing the performance of the test.

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“…For the case of Brownian motion and Brownian bridge on the unit interval [0, 1] the upper and lower bounds for (2) have been investigated in the literatures, see e.g. [4,5] and Hasorva [7] and [8] for boundary non-crossing probabilities. As a by product to our proposed technique, we also get in this work the upper and lower bounds for the probability:…”

Section: Introductionmentioning

confidence: 99%

Estimating the power of lack-of-fit test for the mean of multivariate spatial regression

Somayasa¹

2017

ams

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The purpose of this paper is to establish an estimation of a boundary crossing probability composed by a multivariate centered Gaussian processes and a vector of deterministic trends. Such probability corresponds in statistics to the power function of an asymptotic lack-of-fit test conducted based on the Kolmogorov functional of set-indexed partial (cumulative) sums process of the least squares residuals of multivariate spatial regression. Since the analytical computation of the probability is impossible, we investigate its upper and lower bounds by applying some methods relied on the multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions. Our consideration is mainly for the multivariate set-indexed Brownian pillow. The results are shown not only useful for analyzing the behavior of the test, but also worth for the abstraction and generalization of some existing results toward univariate Gaussian process studied in many literatures.

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Boundary Non-crossings of Brownian Pillow

Cited by 9 publications

References 33 publications

Boundary noncrossings of additive Wiener fields∗

Boundary noncrossings of additive Wiener fields∗

Accessing the Power of Tests Based on Set-Indexed Partial Sums of Multivariate Regression Residuals

Estimating the power of lack-of-fit test for the mean of multivariate spatial regression

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