Abstract
This article investigates the application of depth estimators to crack growth models in construction engineering. Many crack growth models are based on the Paris–Erdogan equation which describes crack growth by a deterministic differential equation. By introducing a stochastic error term, crack growth can be modeled by a non-stationary autoregressive process with Lévy-type errors. A regression depth approach is presented to estimate the drift parameter of the process. We then prove the consistency of the estimator under quite general assumptions on the error distribution. By an extension of the depth notion to simplical depth it is possible to use a degenerated U-statistic and to establish tests for general hypotheses about the drift parameter. Since the statistic asymptotically has a transformed \({\chi_1^2}\) distribution, simple confidence intervals for the drift parameter can be obtained. In the second part, simulations of AR(1) processes with different error distributions are used to examine the quality of the constructed test. Finally we apply the presented method to crack growth experiments. We compare two datasets from independent experiments under different conditions but with the same material. We show that the parameter estimates differ significantly in this case.
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References
Anderson T (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Ann Math Stat 30: 676–687
Basawa I, Mallik A, McCormick W, Taylor R (1989) Bootstrapping explosive autoregressive processes. Ann Stat 17: 1479–1486
Huggins R (1989) The sign test for stochastic processes. Aust J Stat 31: 153–165
Iacus SM (2008) Simulation and inference for stochastic differential equations. With R examples. Springer, New York
Kloeden PE, Platen E, Schurz H (2003) Numerical solution of SDE through computer experiments. Springer, Berlin
Lin L, Chen M (2006) Robust estimating equation based on statistical depth. Stat Pap 47: 263–278
Liu RY (1988) On a notion of simplicial depth. Proc Natl Acad Sci USA 85: 1732–1734
Liu RY (1990) On a notion of data depth based on random simplices. Ann Stat 18: 405–414
Mann H, Wald A (1943) On the statistical treatment of linear stochastic difference equations. Econometrica 11: 173–220
Maurer R, Heeke G (2010) Ermüdungsfestigkeit von Spannstählen aus einer älteren Spannbetonbrücke. TU Dortmund, University. Technical report
Mizera I (2002) On depth and deep points: a calculus. Ann Stat 30: 1681–1736
Müller ChH (2005) Depth estimators and tests based on the likelihood principle with application to regression. J Multivar Anal 95: 153–181
Paulaauskas V, Rachev T (2003) Maximum likelihood estimators in regression models with infinite variance innovations. Stat Pap 44: 47–65
Pook L (2000) Linear elastic fracture mechanics for engineers: theory and application. WIT Press, Southhampton
Rousseeuw PJ, Hubert M (1999) Regression depth. J Am Stat Assoc 94: 388–402
Shevlyakov G, Smirnov P (2011) Robust estimation of the correlation coefficient: an attempt of survey. Aust J Stat 40: 147–156
Stute W, Gründer B (1993) Nonparametric prediction intervals for explosive AR(1)-processes. Nonparametr Stat 2: 155–167
Wellmann R, Harmand P, Müller ChH (2009) Distribution-free tests for polynomial regression based on simplicial depth. J Multivar Anal 100: 622–635
Wellmann R, Müller ChH (2010) Tests for multiple regression based on simplicial depth. J Multivar Anal 101: 824–838
Wellmann R, Müller ChH (2010) Depth notions for orthogonal regression. J Multivar Anal 101: 2358–2371
Witting H, Müller-Funk U (1995) Mathematische Statistik II. Teubner, Stuttgart
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Kustosz, C.P., Müller, C.H. Analysis of crack growth with robust, distribution-free estimators and tests for non-stationary autoregressive processes. Stat Papers 55, 125–140 (2014). https://doi.org/10.1007/s00362-012-0479-5
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DOI: https://doi.org/10.1007/s00362-012-0479-5
Keywords
- Crack growth
- Stochastic differential equation
- Autoregressive process
- Data depth
- Robustness
- Maximum depth estimator
- Simplical depth
- Tests