Analysis of crack growth with robust, distribution-free estimators and tests for non-stationary autoregressive processes

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.

[1]  T. W. Anderson.  On Asymptotic Distributions of Estimates of Parameters of Stochastic Difference Equations , 1959 .

[2]  Pavel O. Smirnov,et al.  Robust Estimation of the Correlation Coefficient: An Attempt of Survey , 2011 .

[3]  A. Wald,et al.  On the Statistical Treatment of Linear Stochastic Difference Equations , 1943 .

[4]  Svetlozar T. Rachev,et al.  Maximum likelihood estimators in regression models with infinite variance innovations , 2003 .

[5]  R. Y. Liu,et al.  On a notion of simplicial depth. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Christine H. Müller,et al.  Depth notions for orthogonal regression , 2010, J. Multivar. Anal..

[7]  L. P. Pook,et al.  Linear Elastic Fracture Mechanics for Engineers: Theory and Applications , 2000 .

[8]  Nonparametric prediction intervals for explosive ar(1)-processes , 1993 .

[9]  Stefano M. Iacus,et al.  Simulation and Inference for Stochastic Differential Equations: With R Examples , 2008 .

[10]  I. Mizera.  On depth and deep points: a calculus , 2002 .

[11]  Regina Y. Liu,et al.  Regression depth. Commentaries. Rejoinder , 1999 .

[12]  Minghua Chen,et al.  Robust estimating equation based on statistical depth , 2006 .

[13]  Christine H. Müller,et al.  Distribution-free tests for polynomial regression based on simplicial depth , 2009, J. Multivar. Anal..

[14]  R. Huggins.  THE SIGN TEST FOR STOCHASTIC PROCESSES , 1989 .

[15]  Hermann Witting,et al.  Mathematische Statistik II , 1985 .

[16]  M ChristineH..  Depth estimators and tests based on the likelihood principle with application to regression , 2004 .

[17]  P. Kloeden,et al.  Numerical Solution of Sde Through Computer Experiments , 1993 .

[18]  Regina Y. Liu.  On a Notion of Data Depth Based on Random Simplices , 1990 .

[19]  Christine H. Müller,et al.  Tests for multiple regression based on simplicial depth , 2010, J. Multivar. Anal..

[20]  R. Taylor,et al.  Bootstrapping explosive autoregressive processes , 1989 .