Simplified simplicial depth for regression and autoregressive growth processes

Kustosz, Christoph P.; Müller, Christine H.; Wendler, Martin

doi:10.1016/j.jspi.2016.01.005

Cited by 9 publications

(9 citation statements)

References 35 publications

(28 reference statements)

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“…It holds for all cases where depth of a two‐dimensional parameter at three data points is given by alternating signs of the three residuals. This holds for several other models as shown in Kustosz et al ().…”

Section: Asymptotic Distribution Of the Simplicial Depthsupporting

confidence: 72%

“…The simplicial depth is a U‐statistic. In the AR(1) model , it becomes

d_{S} (θ, z_{*}) = \frac{1}{(\frac{0.0pt}{N})} \sum_{1 \leq n_{1} < n_{2} < n_{3} \leq N} 1 {d_{T} (θ, (z_{n_{1}}, z_{n_{2}}, z_{n_{3}})) > 0} .

If the regressors y n − 1 satisfy

y_{n_{1} - 1} < y_{n_{2} - 1} < y_{n_{3} - 1}

for n 1 < n 2 < n 3 , then

d_{T} (θ, (z_{n_{1}}, z_{n_{2}}, z_{n_{3}})) > 0

if and only if the residuals

r_{n_{1}}, r_{n_{2}}, r_{n_{3}}

have alternating signs or at least one of them is zero (Kustosz et al ., ). Since Y n is almost surely strictly increasing by assumption, we can always assume

y_{n_{1} - 1} < y_{n_{2} - 1} < y_{n_{3} - 1}

for n 1 < n 2 < n 3 without loss of generality.…”

Section: Simplicial Depth For the Ar(1) Modelmentioning

confidence: 97%

“…To define simplicial depth for autoregression, it is useful to write the sample in pairs, that is, the sample is given by´ D .´1; : : : ;´N / > where´n D .y n ; y n 1 / > . Then, simplicial depth of a p-dimensional parameter Â 2 R p in the sample´ is in general (Müller, 2005) If the regressors y n 1 satisfy y n1 1 < y n2 1 < y n3 1 for n 1 < n 2 < n 3 , then d T .Â; .´n 1 ;´n 2 ;´n 3 // > 0 if and only if the residuals r n1 ; r n2 ; r n3 have alternating signs or at least one of them 766 C. P. KUSTOSZ, A. LEUCHT AND C. H. MÜLLER is zero (Kustosz et al, 2016). Since Y n is almost surely strictly increasing by assumption, we can always assume y n1 1 < y n2 1 < y n3 1 for n 1 < n 2 < n 3 without loss of generality.…”

Section: Simplicial Depth For the Ar(1) Modelmentioning

confidence: 99%

“…The Supporting Information shows a very similar behaviour of the three tests for errors with a Fréchet distribution. Moreover, three simplifications of the simplicial depth test, proposed by Kustosz et al (), are compared with the simplicial depth test, the sign test and the OLS test there. However, these simplifications show a worse behaviour than the simplicial depth test, although they outperform the OLS test under nonnormal error distributions and do not have the drawback of the sign test, which leads to an unbounded area of very small power.…”

Section: Simulation Of the Power Of The Testmentioning

confidence: 99%

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Tests Based on Simplicial Depth for AR(1) Models With Explosion

Kustosz

Leucht

Müller

2016

Journal Time Series Analysis

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We propose an outlier robust and distributions-free test for the explosive AR(1) model with intercept based on simplicial depth. In this model, simplicial depth reduces to counting the cases where three residuals have alternating signs. Using this, it is shown that the asymptotic distribution of the test statistic is given by an integrated two-dimensional Gaussian process. Conditions for the consistency of the test are given and the power of the test at finite samples is compared with five alternative tests, using errors with normal distribution, contaminated normal distribution, and Fréchet distribution in a simulation study. The comparisons show that the new test outperforms all other tests in the case of skewed errors and outliers. Although here we deal with the AR(1) model with intercept only, the asymptotic results hold for any simplicial depth which reduces to alternating signs of three residuals.

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Section: Asymptotic Distribution Of the Simplicial Depthsupporting

confidence: 72%

“…The simplicial depth is a U‐statistic. In the AR(1) model , it becomes

d_{S} (θ, z_{*}) = \frac{1}{(\frac{0.0pt}{N})} \sum_{1 \leq n_{1} < n_{2} < n_{3} \leq N} 1 {d_{T} (θ, (z_{n_{1}}, z_{n_{2}}, z_{n_{3}})) > 0} .

If the regressors y n − 1 satisfy

y_{n_{1} - 1} < y_{n_{2} - 1} < y_{n_{3} - 1}

for n 1 < n 2 < n 3 , then

d_{T} (θ, (z_{n_{1}}, z_{n_{2}}, z_{n_{3}})) > 0

if and only if the residuals

r_{n_{1}}, r_{n_{2}}, r_{n_{3}}

have alternating signs or at least one of them is zero (Kustosz et al ., ). Since Y n is almost surely strictly increasing by assumption, we can always assume

y_{n_{1} - 1} < y_{n_{2} - 1} < y_{n_{3} - 1}

for n 1 < n 2 < n 3 without loss of generality.…”

Section: Simplicial Depth For the Ar(1) Modelmentioning

confidence: 97%

Section: Simplicial Depth For the Ar(1) Modelmentioning

confidence: 99%

Section: Simulation Of the Power Of The Testmentioning

confidence: 99%

See 2 more Smart Citations

Tests Based on Simplicial Depth for AR(1) Models With Explosion

Kustosz

Leucht

Müller

2016

Journal Time Series Analysis

Self Cite

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show abstract

“…Hence, they are outlier-robust. However, previous results 25,26 indicate that tests based on (p + 1)-sign depth with p ≥ 2 are much more powerful than the classical sign test.…”

Section: New Confidence Sets For the Parameter Based On Sign Depthmentioning

confidence: 92%

Prediction intervals for load‐sharing systems in accelerated life testing

Leckey

Müller

Szugat

et al. 2020

Quality & Reliability Eng

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Based on accelerated lifetime experiments, we consider the problem of constructing prediction intervals for the time point at which a given number of components of a load-sharing system fails. Our research is motivated by lab experiments with prestressed concrete beams where the tension wires fail successively. Due to an audible noise when breaking, the time points of failure could be determined exactly by acoustic measurements. Under the assumption of equal load sharing between the tension wires, we present a model for the failure times based on a birth process. We provide a model check based on a Q-Q plot including a simulated simultaneous confidence band and four simulation-free prediction methods. Three of the prediction methods are given by confidence sets where two of them are based on classical tests and the third is based on a new outlier-robust test using sign depth. The fourth method uses the implicit function theorem and the-method to get prediction intervals without confidence sets for the unknown parameter. We compare these methods by a leave-one-out analysis of the data on prestressed concrete beams. Moreover, a simulation study is performed to discuss advantages and drawbacks of the individual methods.

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Simple powerful robust tests based on sign depth

et al. 2022

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Up to now, powerful outlier robust tests for linear models are based on M-estimators and are quite complicated. On the other hand, the simple robust classical sign test usually provides very bad power for certain alternatives. We present a generalization of the sign test which is similarly easy to comprehend but much more powerful. It is based on K-sign depth, shortly denoted by K-depth. These so-called K-depth tests are motivated by simplicial regression depth, but are not restricted to regression problems. They can be applied as soon as the true model leads to independent residuals with median equal to zero. Moreover, general hypotheses on the unknown parameter vector can be tested. While the 2-depth test, i.e. the K-depth test for $$K = 2$$ K = 2 , is equivalent to the classical sign test, K-depth test with $$K\ge 3$$ K ≥ 3 turn out to be much more powerful in many applications. A drawback of the K-depth test is its fairly high computational effort when implemented naively. However, we show how this inherent computational complexity can be reduced. In order to see why K-depth tests with $$K\ge 3$$ K ≥ 3 are more powerful than the classical sign test, we discuss the asymptotic behavior of its test statistic for residual vectors with only few sign changes, which is in particular the case for some alternatives the classical sign test cannot reject. In contrast, we also consider residual vectors with alternating signs, representing models that fit the data very well. Finally, we demonstrate the good power of the K-depth tests for some examples including high-dimensional multiple regression.

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Simplified simplicial depth for regression and autoregressive growth processes

Cited by 9 publications

References 35 publications

Tests Based on Simplicial Depth for AR(1) Models With Explosion

Tests Based on Simplicial Depth for AR(1) Models With Explosion

Prediction intervals for load‐sharing systems in accelerated life testing

Simple powerful robust tests based on sign depth

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