Testing for Lack of Fit in Inverse Regression—with Applications to Biophotonic Imaging

Bissantz, Nicolai; Claeskens, Gerda; Holzmann, Hajo; Munk, Axel

doi:10.1111/j.1467-9868.2008.00670.x

Cited by 21 publications

(23 citation statements)

References 42 publications

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“…We refer to Aerts et al (2000) and Bissantz et al (2009) construction that could be of particular interest would be to combine score statistics instead.…”

Section: Discussion and Extensionsmentioning

confidence: 99%

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Order selection tests with multiply imputed data

Consentino¹,

Claeskens²

2010

Computational Statistics & Data Analysis

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We develop nonparametric tests for the null hypothesis that a function has a prescribed form, to apply to data sets with missing observations. Omnibus nonparametric tests do not need to specify a particular alternative parametric form, and have power against a large range of alternatives, the order selection tests that we study are one example. We extend such order selection tests to be applicable in the context of missing data. In particular, we consider likelihood-based order selection tests for multiplyimputed data. A simulation study and data analysis illustrate the performance of the tests. A model selection method in the style of Akaike's information criterion for multiply imputed datasets results along the same lines.

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“…We refer to Aerts et al (2000) and Bissantz et al (2009) construction that could be of particular interest would be to combine score statistics instead.…”

Section: Discussion and Extensionsmentioning

confidence: 99%

“…Recently, these tests have been studied for inverse regression problems by Bissantz et al (2009). Test statistics can be based on likelihood ratio, Wald or score statistics.…”

Section: Introductionmentioning

confidence: 99%

Order selection tests with multiply imputed data

Consentino¹,

Claeskens²

2010

Computational Statistics & Data Analysis

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“…The signalto-noise ratio of the images is S/N ≈ 20 and S/N ≈ 14 for bead1 and bead2, respectively. For a more detailed discussion of the HeLa cell data we refer to Bissantz et al [7].…”

Section: True Functionmentioning

confidence: 99%

Testing for Image Symmetries—With Application to Confocal Microscopy

Bissantz

Holzmann

Pawlak

2009

IEEE Trans. Inform. Theory

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Statistical tests are introduced for checking whether an image function f (x, y) defined on the unit disc D = {(x, y) : x 2 + y 2 ≤ 1} is invariant under certain symmetry transformations of D, given that discrete and noisy data are observed. We consider invariance under reflections or under rotations by rational angles, as well as joint invariance under both reflections and rotations. Furthermore, we propose a test for rotational invariance of f (x, y), i.e., for checking whether f (x, y), after transformation to polar coordinates, only depends on the radius and not on the angle. These symmetry relations can be naturally expressed as restrictions for the Zernike moments of the image function f (x, y), i.e., the Fourier coefficients with respect to the Zernike orthogonal basis. Therefore, our test statistics are based on checking whether the estimated Zernike coefficients approximately satisfy those restrictions. This is carried out by forming the L 2 distance between the image function and its transformed version obtained by some symmetry transformation. We derive the asymptotic distribution of the test statistics under both the hypothesis of symmetry as well as under fixed alternatives. Furthermore, we investigate the quality of the asymptotic approximations via simulation studies. The usefulness our theory is verified by examining an important problem in confocal microscopy, i.e., we investigate possible imprecise alignments in the optical path of the microscope. For optical systems with rotational symmetry, the theoretical point-spread-function (PSF) is reflection symmetric with respect to two orthogonal axes, and rotationally invariant if the detector plane matches the optical plane of the microscope. We use our tests to investigate whether the required symmetries can indeed be detected in the empirical PSF.

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“…Remark that the SVD can explicitly computed for a large class of inverse problems, e.g. tomography (see [19]), deconvolution (see [20]) or Biophotonic imaging (see [5]). …”

Section: Introductionmentioning

confidence: 99%

Risk hull method for spectral regularization in linear statistical inverse problems

Marteau

2010

ESAIM: PS

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Abstract. We consider in this paper the statistical linear inverse problem Y = Af + ξ where A denotes a compact operator, a noise level and ξ a stochastic noise. The unknown function f has to be recovered from the indirect measurement Y . We are interested in the following approach: given a family of estimators, we want to select the best possible one. In this context, the unbiased risk estimation (URE) method is rather popular. Nevertheless, it is also very unstable. Recently, Cavalier and Golubev (2006) introduced the risk hull minimization (RHM) method. It significantly improves the performances of the standard URE procedure. However, it only concerns projection rules. Using recent developments on ordered processes, we prove in this paper that it can be extended to a large class of linear estimators.Mathematics Subject Classification. 62G05, 62G20.

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Testing for Lack of Fit in Inverse Regression—with Applications to Biophotonic Imaging

Cited by 21 publications

References 42 publications

Order selection tests with multiply imputed data

Order selection tests with multiply imputed data

Testing for Image Symmetries—With Application to Confocal Microscopy

Risk hull method for spectral regularization in linear statistical inverse problems

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