Mathematische Statistik II

a b s t r a c tA general approach for developing distribution-free tests for general linear models based on simplicial depth is presented. In most relevant cases, the test statistic is a degenerated U-statistic so that the spectral decomposition of the conditional expectation of the kernel function is needed to derive the asymptotic distribution. A general formula for this conditional expectation is derived. Then it is shown how this general formula can be specified for polynomial regression. Based on the specified form, the spectral decomposition and thus the asymptotic distribution is derived for polynomial regression of arbitrary degree. The power of the new test is compared via simulation with other tests. An application on cubic regression demonstrates the applicability of the new tests and in particular their outlier robustness.

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“…We have namely the following result (see e.g. [6], p. 79, 80, 90, [21], p. 650). If the reduced normalized kernel function…”

Section: Introductionmentioning

confidence: 87%

Distribution-free tests for polynomial regression based on simplicial depth

Wellmann

Harmand

Müller

2009

Journal of Multivariate Analysis

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“…We may draw on the asympotic theory of order statistics to find some A ∈ F, P(A) = 1, such that holds for every ω ∈ A and any j ∈ N, where [N α j + 1] denotes the largest l ∈ N with l ≤ N α j + 1 (cf. Witting and Müller-Funk (1995), Satz 7.108, Satz 7.120). Now let us fix ω ∈ A.…”

Section: Proofmentioning

confidence: 97%

A Microeconomic Explanation of the EPK Paradox

Härdle

Krätschmer²,

Moro

2009

SSRN Journal

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Supported by several recent investigations the empirical pricing kernel paradox might be considered as a stylized fact. In Chabi-Yo et al. (2008) simulation studies have been presented which suggest that this paradox might be caused by regime switching of stock prices in financial markets. Alternatively, we want to emphasize a microeconomic view. Based on an economic model with state dependent utilities for the financial investors we succeed in explaining the paradox by changes of the risk attitudes. Theoretically, the change behaviour is compressed by the pricing kernels. As a starting point for empirical insights we shall develop and investigate inverse problems in terms of data fits for estimated basic values of the pricing kernel.

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“…(b) Note that it is only necessary to check (33) and (34) there are lots of proofs of central limit theorems for L-statistics with discrete and continuous weights under different assumptions, see for example Mason and Shorack (1992), Stigler (1974), Shorack and Wellner (1986, p. 664, Theorem 1), Witting andMüller-Funk (1995, Theorem 7.210), or van der Vaart (1998, Theorem 22.3). The asymptotic variance of the limit normal distribution turns out to equal our information bound (26) for the L-functionals.…”

Section: Theorem 3 Suppose Thatmentioning

confidence: 99%

Information bounds for nonparametric estimators of L-functionals and survival functionals under censored data

Janssen

Knoch

2015

Metrika

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In the present paper we derive lower asymptotic information bounds of Cramér-Rao type for estimators of nonparametric statistical functionals. The results are based on dense differentiability and dense regularity concepts which lead to weak assumptions. As explicit examples L-estimators are treated. In addition a new rapid method for the treatment of survival functionals under randomly right censored data is presented. For instance, for the famous Kaplan-Meier and Nelson-Aalen estimators, our information bound is just the lower bound obtained earlier in the literature.

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Mathematische Statistik II

Cited by 53 publications

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Distribution-free tests for polynomial regression based on simplicial depth

Distribution-free tests for polynomial regression based on simplicial depth

A Microeconomic Explanation of the EPK Paradox

Information bounds for nonparametric estimators of L-functionals and survival functionals under censored data

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