Abstract
In the present paper, we are mainly concerned with the degradation mechanism that arises in fatigue crack growth (FCG). The crack evolution mechanism is modeled by a first order stochastic differential system, composed by a deterministic FCG equation perturbed by a stochastic process. The main purpose is to investigate the estimators of the model parameters and establish some asymptotic properties by transforming the initial equation into a regression model. To this purpose, least squares estimation (LSE) is applied in the framework of the errors being martingale differences. The parametric conditional LSE are proved to be consistent. Our results are obtained in the general case and specified for the linear case with focus on a particular application model derived from the most widely used FCG law in fracture mechanics, i.e., the Paris model. We derive the asymptotic normality of the LSE for the nonlinear case with discussion for the linear case. Finally, we provide a numerical example to illustrate the performance of the proposed methodology, on a particular version of the stochastic model.
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References
Anderson TW, Taylor JB (1979) Strong consistency of least squares estimates in dynamic models. Ann. Stat. 7(3):484–489
Banerjee P, Karpenko O, Udpa L, Haq M, Deng Y (2018) Prediction of impact-damage growth in GFRP plates using particle filtering algorithm. Compos. Struct. 194:527–536
Bellec PC (2018) Sharp oracle inequalities for least squares estimators in shape restricted regression. Ann. Stat. 46(2):745–780
Abdessalem AB, Azaïs R, Touzet-Cortina M, Gégout-Petit A, Puiggali M (2016) Stochastic modelling and prediction of fatigue crack propagation using piecewise-deterministic markov processes. Proc. Inst. Mech. Eng. Part O: J. Risk Reliab. 230(4):405–416
Bickel PJ, Klaassen CAJ, Ritov Y, Wellner JA (1998) Efficient and Adaptive Estimation for Semiparametric Models. Springer, New York. Reprint of the 1993 original
Bindele HF (2015) The signed-rank estimator for nonlinear regression with responses missing at random. Electron. J. Stat. 9(1):1424–1448
Chen F, Zou B, Chen N (2018) The consistency of least-square regularized regression with negative association sequence. Int. J. Wavelets Multiresolut. Inf. Process. 16(3):1850019–1850020
Chen X (2012) Asymptotic properties for estimates of nonparametric regression model with martingale difference errors. Statistics 46(5):687–696
Chen Z, Wang H, Wang X (2016) The consistency for the estimator of nonparametric regression model based on martingale difference errors. Stat. Pap. 57(2):451–469
Cheng R (2017) Non-standard Parametric Statistical Inference. Oxford University Press, Oxford
Chiquet J, Limnios N (2008) A method to compute the transition function of a piecewise deterministic Markov process with application to reliability. Stat. Probab. Lett. 78(12):1397–1403
Chiquet J, Limnios N (2013) Dynamical systems with semi-Markovian perturbations and their use in structural reliability. In: Stochastic Reliability and Maintenance Modeling, Volume 9 of Springer Series of Reliability and Engineering, pp 191–218. Springer, London
Chiquet J, Limnios N, Eid M (2009) Piecewise deterministic markov processes applied to fatigue crack growth modelling. J. Stat. Plan. Inference 139(5):1657–1667
Christopeit N, Helmes K (1980) Strong consistency of least squares estimators in linear regression models. Ann. Stat. 8(4):778–788
Cinlar E (1969) Markov renewal theory. Adv. Appl. Probab. 1(2):123–187
Collipriest JE (1972) An experimentalist’s view of the surface flaw problem. In: Swedlow EJL (ed) Physical Problems and Computational Solutions. American Society of Mechanical Engineers, New York, pp 43–62
Davis MHA (1984) Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. Roy. Stat. Soc. Ser. B 46(3):353–388
Davis MHA (1993) Markov Models and Optimization. Volume 49 of Monographs on Statistics and Applied Probability. Chapman & Hall, London
Delgado MA (1992) Semiparametric generalized least squares in the multivariate nonlinear regression model. Econ. Theory 8(2):203–222
Donaldson JR, Schnabel RB (1987) Computational experience with confidence regions and confidence intervals for nonlinear least squares. Technometrics 29(1):67–82
Draper NR, Smith H (1998) Applied Regression Analysis, 3rd edn. Wiley, Hoboken
Eicker F (1963) Über die konsistenz von parameterschätzfunktionen für ein gemischtes zeitreihen-regressionsmodell. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1(5):456–477
Godambe VP, Heyde CC (1987) Quasi-likelihood and optimal estimation. Int. Stat. Rev. 55:231–244
Grenander U, Rosenblatt M (1957) Statistical Analysis of Stationary Time Series. Wiley, New York
Grigoriev Y, Ivanov AV (1993) Asymptotic expansions for quadratic functionals of the least squares estimator of a nonlinear regression parameter. Math. Methods Stat. 2(4):269–294
Hall P, Heyde CC (1980) Martingale Limit Theory and Its Application. Probability and Mathematical Statistics. Academic Press Inc, New York
Heyde CC (1997) Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation. Springer, New York
Howard R (1971) Dynamic Probabilistic Systems: Volume I: Markov Models. Series in Decision and Control. Wiley, Hoboken
Huber PJ (1973) Robust regression: asymptotics, conjectures and Monte Carlo. Ann. Stat. 1:799–821
Ibragimov R, Phillips PCB (2008) Regression asymptotics using martingale convergence methods. Econ. Theory 24(4):888–947
Jacob C (2010) Conditional least squares estimation in nonstationary nonlinear stochastic regression models. Ann. Stat. 38(1):566–597
Jacobsen M (2006) Point process theory and applications: marked point and piecewise deterministic processes. In: Probability and its Applications, Birkhäuser Boston Inc, Boston
Jennrich RI (1969) Asymptotic properties of non-linear least squares estimators. Ann. Math. Stat. 40(2):633–643
Koroliuk VS, Limnios N (2005) Stochastic Systems in Merging Phase Space. World Scientific, London
Lalam N, Jacob C (2004) Estimation of the offspring mean in a supercritical or near- critical size-dependent branching process. Adv. Appl. Probab. 36:582–601
Lai TL (1994) Asymptotic properties of nonlinear least squares estimates in stochastic regression models. Ann. Stat. 22(4):1917–1930
Lai TL, Robbins H (1981) Consistency and asymptotic efficiency of slope estimates in stochastic approximation schemes. Probab. Theory Relat. Fields 56(3):329–360
Lai TL, Wei CZ (1982) Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann. Stat. 10(1):154–166
Lai TL, Robbins H, Wei CZ (1978) Strong consistency of least squares estimates in multiple regression. Proc. Nat. Acad. Sci. U.S.A. 75(7):3034–3036
Lai TL, Robbins H, Wei CZ (1979) Strong consistency of least squares estimates in multiple regression. II. J. Multivariate Anal. 9(3):343–361
Lehmann EL, Casella G (1998) Theory of Point Estimation, 2nd edn. Springer Texts in Statistics. Springer, New York
Lehmann EL, Romano JP (2005) Testing Statistical Hypotheses, 3rd edn. Springer Texts in Statistics. Springer, New York
Li D, Tjøstheim D, Gao J (2016) Estimation in nonlinear regression with Harris recurrent Markov chains. Ann. Stat. 44(5):1957–1987
Limnios N, Oprişan G (2001) Semi-Markov processes and reliability. Statistics for Industry and Technology, Birkhäuser Boston Inc, Boston, MA
Lin YK, Yang JN (1985) A stochastic theory of fatigue crack propagation. Am. Inst. Aeronaut. Astron. 23(1):117–124
Lindsey JK (1996) Parametric Statistical Inference. Oxford Science Publications, Oxford University Press, New York
Liu X, Ouyang A, Yun Z (2018) Fuzzy weighted least squares support vector regression with data reduction for nonlinear system modeling. Math. Probl. Eng. Art. ID 7387650, 13
Malinvaud E (1970) The consistency of nonlinear regressions. Ann. Math. Stat. 41(3):956–969
Malinvaud E (1980) Statistical methods of econometrics, Volume 6 of Studies in Mathematical and Managerial Economics. North-Holland Publishing Co., Amsterdam, third edition. Translated from the French by Anne Silvey
Miller S, Startz R (2019) Feasible generalized least squares using support vector regression. Econ. Lett. 175:28–31
Nelson PI (1980) A note on strong consistency of least squares estimators in regression models with martingale difference errors. Ann. Stat. 8(5):1057–1064
Øksendal B (2003) Stochastic Differential Equations: An Introduction with Applications. Hochschultext/Universitext. US, Government Printing Office
Papamichail CA, Bouzebda S, Limnios N (2016) Reliability Calculus on Crack Propagation Problem with a Markov Renewal Process. Springer, New York, pp 343–378
Paris PC, Erdogan F (1963) A critical analysis of crack propagation laws. J. Fluids Eng. 85:528–533
Peng Z, Ying QG, Liao B, Ren XF (2018) Study of fatigue crack propagation behaviour for dual-phase X80 pipeline steel. Ironmak. Steelmak. 45(7):635–640
Pfanzagl J (1994) Parametric statistical theory. De Gruyter Textbook. Walter de Gruyter & Co., Berlin. With the assistance of R. Hamböker
Prakasa Rao BLS (1984) The rate of convergence of the least squares estimator in a nonlinear regression model with dependent errors. J. Multivariate Anal. 14(3):315–322
Prakasa Rao BLS (1986) Weak convergence of least squares process in the smooth case. Statistics 17(4):505–516
Pronzato L (2009) Asymptotic properties of nonlinear estimates in stochastic models with finite design space. Stat. Probab. Lett. 79(21):2307–2313
Pyke R (1961) Markov renewal processes: definitions and preliminary properties. In: The Annals of Mathematical Statistics, pp. 1231–1242
Shklyar S (2018) Consistency of the total least squares estimator in the linear errors-in-variables regression. Mod. Stoch. Theory Appl. 5(3):247–295
Shurenkov VM (1984) On the theory of markov renewal. Theory Probab. Appl. 29(2):247–265
Skouras K (2000) Strong consistency in nonlinear stochastic regression models. Ann. Stat. 28(3):871–879
Sobczyk K (1993) Stochastic approach to fatigue: experiments, modelling, and reliability estimation. Springer, CISM International Centre for Mechanical Sciences Series
Sobczyk K, Spencer B (2012) Random Fatigue: From Data to Theory. Elsevier, Amsterdam
Spencer BF, J (1993) Stochastic diffusion models for fatigue crack growth and reliability estimation. In K. Sobczyk, (Ed.), Stochastic Approach to Fatigue, Volume 334 of International Centre for Mechanical Sciences, pp 185–241, Springer, Vienna
Tong H, Ng M (2018) Analysis of regularized least squares for functional linear regression model. J. Complex. 49:85–94
van der Vaart AW (1998) Asymptotic Statistics. Volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge
Virkler DA, Hillberry BM, Goel PK (1978) The statistical nature of fatigue crack propagation. In: Technical report, School of Mechanical Engineering Purdue University West Lafayette, Indiana
Šidák Z (1967) Rectangular confidence regions for the means of multivariate normal distributions. J. Am. Stat. Assoc. 62(318):626–633
Wang J (1996) Asymptotics of least-squares estimators for constrained nonlinear regression. Ann. Stat. 24(3):1316–1326
Wang X, Deng X, Hu S (2018) On consistency of the weighted least squares estimators in a semiparametric regression model. Metrika 81(7):797–820
Wedderburn RWM (1974) Quasi-likelihood functions, generalized linear models, and the gauss-newton method. Biometrika 61(3):439–447
Wu C-F (1981) Asymptotic theory of nonlinear least squares estimation. Ann. Stat. 9(3):501–513
Zhang S, Miao Y, Xu X, Gao Q (2018) Limit behaviors of the estimator of nonparametric regression model based on martingale difference errors. J. Korean Stat. Soc. 47(4):537–547
Zhou X-C, Lin J-G (2012) A wavelet estimator in a nonparametric regression model with repeated measurements under martingale difference error’s structure. Statist. Probab. Lett. 82(11):1914–1922
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A Proofs of Sections 3, 5
A Proofs of Sections 3, 5
This section is devoted to the detailed proofs of our results. The previously displayed notation continue to be used in the sequel.
1.1 Proof of Proposition 3.5
Similarly to Theorem 1 of [36], it suffices to show
where
with
This implies that
By application of Theorem 1 of [74] or Proposition 3.1 of [31], we should show that, for all \(\varvec{\theta } \ne \varvec{\theta _0}\),
which follows from Assumption A.3. It suffices to show that
This result exactly follows from the strong law of large numbers for submartingales (Proposition 5.1 of [31]), applied to \(d_{k}(\varvec{\theta }), \Vert \varvec{\theta }-\varvec{\theta }_{0}\Vert \ge \delta ,\) by using the assumptions A.1-3. Thus the proof is complete. \(\square\)
1.2 Proof of Theorem 5.2
Following [74] and the proof of Proposition 6.1 of [31], we infer from the first order Taylor series expansion of \(\mathbf { S}_{n}^{\prime }(\widehat{\varvec{\theta }}_{n})\) at \(\varvec{\theta }_{0}\) that
Since \(\widehat{\varvec{\theta }}_{n}\) is an interior point of \(\varvec{\Theta }\) eventually \(\mathbf { S}_{n}^{\prime }(\widehat{\varvec{\theta }}_{n})=0\). With \({\mathbf {S}}_{n}^{\prime \prime }(\cdot )\) invertible in a neighborhood of \(\varvec{\theta }_{0}\), assumption B.1.iii, Eq. (A.2) implies
and also
Hence for any \(p\times p\) matrix \(\Psi _{n}\), such that (A.1) in Jacob [31] is satisfied, we have
and also
From
we obtain its first derivative
and then, the second derivative
So, from (A.4), \(\mathbf { S}_{n}^{\prime }({\varvec{\theta }})\) at \(\varvec{\theta }_{0}\) takes the value
Notice that the following terms
-
1.
\(\displaystyle \left[ \sum _{k=1}^{n}\mathbf { f}^{\prime }_{k}(\widetilde{\varvec{\theta }}_{n})\mathbf { f}^{\prime }_{k}(\widetilde{\varvec{\theta }}_{n})^{\top }\right] {\mathbf {M}}_{n} {\mathop \rightarrow \limits ^{{\mathbb {P}}}} {\mathbf {I}} ~~\text{ according } \text{ to } \text{ assumption } \text{ B.1.(iii) },\)
-
2.
\(\displaystyle \left[ \sum _{k=1}^{n}\eta _{k}\mathbf { f}^{\prime \prime }_{k}(\widetilde{\varvec{\theta }}_{n})\right] {\mathbf {M}}_{n} {\mathop \rightarrow \limits ^{{\mathbb {P}}}} 0 , ~~\text{ according } \text{ to } \text{ assumption } \text{ B.1.(i) },\)
-
3.
\(\displaystyle \left[ \sum _{k=1}^{n}\Big (f_{k}(\varvec{ \theta }_{0})-f_{k}(\widetilde{\varvec{\theta }}_{n})\Big )\mathbf { f}^{\prime \prime }_{k}(\widetilde{\varvec{\theta }}_{n})\right] {\mathbf {M}}_{n} {\mathop \rightarrow \limits ^{{\mathbb {P}}}} 0 ~~\text{ according } \text{ to } \text{ assumption } \text{ B.1.(ii) }.\)
Let us define the matrix \({\varvec{K}}_n\) by
We infer that we have
An application of Slutsky’s theorem in the Eq. (A.3), since, as \(n\rightarrow \infty\),
implies that
Then we have
and
which from the assumption B.2 that
exists in distribution, this suffices to complete the proof of Theorem 5.2. \(\square\)
1.3 Proof of Theorem 6.3
This demonstration follows closely the proof of Theorem 2 of Lai [36]. By (A.4) and since \(\mathbf { S}_{n}^{\prime }(\widehat{\varvec{\theta }}_{n})=0\), we have
An application of the mean value theorem implies readily that
which gives
and so
Again from the mean value theorem, one can see that
We infer that we have
It follows that
By multiplying by \(\varvec{\Lambda }_{n}^{1/2}\), we obtain
Then, we evaluate the preceding terms one by one, for all large n. It will be shown that all terms tend to 0 in probability, except for the first one, which appears in the central limit theorem, (Eq. 6.9). We work with suitably chosen Hilbert spaces H and by making use of martingale central limit theorems and probability bounds for H-valued martingales. Recall that \(\{\eta _k, {\mathscr {F}}_{k-1}, k\ge 1 \}\) are martingale differences and \(\{\mathbf { f}^{\prime \prime }_{k}(\varvec{\theta })-\mathbf { f}^{\prime \prime }_{k}(\varvec{\theta }_0)\}\) are \({\mathscr {F}}_{k-1}\) measurable H-valued random variables. The last term of the previous equation gives
with
where we have used assumption C.4 for the last equation.
In more detail now, it is important to see here that the sup-norm of
gets bounded by its H-norm, with the use of Cauchy-Schwarz inequality. Since
this implies
Making use of the Cauchy–Schwarz inequality, we infer that
We obtain
For nonrandom constant \(c_k\) sufficiently large so that
By Borel-Cantelli lemma, we have
We readily infer that
provided that
Hence, from (A.12) and (A.13), it follows
Again, an application of Cauchy-Schwarz inequality gives
We now treat the following term
Let us treat the following term,
Once more, an application of Cauchy–Schwarz inequality implies that
Making use of the Assumption B.1.iii, we infer that
So, it comes out from the above that
or
Following Theorem 2 of [36], the martingale central limit theorem of [26] (in chaptrer 3 and chapter 6) is applied, since the following conditions are satisfied:
1.4 Statement D
-
1.
\(\max \left\| \varvec{\Lambda }_{n}^{-1/2}\eta _k\mathbf { f}^{\prime }_{k}(\varvec{\theta }_{0}) \right\| {\mathop \rightarrow \limits ^{{\mathbb {P}}}} 0\) , and
$$\begin{aligned} {\left\{ \begin{array}{ll} \max \left\| \varvec{\Lambda }_{n}^{-1/2}{\mathbf {f}}^{\prime }_{k}(\varvec{\theta }_{0}) \right\| {\mathop \rightarrow \limits ^{{\mathbb {P}}}} 0, \qquad \text{ from } \text{ assumption } \text{ C.2 } \\ \qquad \text{ and }\\ \sup _k {\mathbb {E}} \big (\eta _k^2 | {\mathscr {F}}_{k-1}\big ) < \infty .\\ \end{array}\right. } \end{aligned}$$ -
2.
Also
$$\begin{aligned} \sum _{k=1}^{n} \varvec{\Lambda }_{n}^{-1/2} \eta _k {\mathbf {f}}_{k}^{\prime }(\varvec{ \theta }_{0}) \sum _{k=1}^{n} \left[ \varvec{\Lambda }_{n}^{-1/2} \eta _k {\mathbf {f}}_{k}^{\prime }(\varvec{ \theta }_{0})\right] ^{\top } {\mathop \rightarrow \limits ^{{\mathbb {P}}}} \sigma ^2 \varvec{\Lambda }_{n}^{-1/2} \sum _{k=1}^{n} {\mathbf {f}}_{k}^{\prime }(\varvec{ \theta }_{0}) {\mathbf {f}}_{k}^{\prime }(\varvec{ \theta }_{0})^{\top } {\varvec{\Lambda }_{n}^{-1/2}}^{\top } . \end{aligned}$$By using the fact that
$$\begin{aligned} \varvec{\Lambda }_{n}^{-1/2}\left( \sum _{k=1}^{n} {\mathbf {f}}_{k}^{\prime }(\varvec{ \theta }_{0}) {\mathbf {f}}_{k}^{\prime }(\varvec{ \theta }_{0})^{\top }\right) ^{1/2} {\mathop \rightarrow \limits ^{{\mathbb {P}}}} {\mathbf {I}}, \end{aligned}$$we readily obtain that
$$\begin{aligned} \sum _{k=1}^{n} \varvec{\Lambda }_{n}^{-1/2} \eta _k {\mathbf {f}}_{k}^{\prime }(\varvec{ \theta }_{0}) \sum _{k=1}^{n} \left[ \varvec{\Lambda }_{n}^{-1/2} \eta _k {\mathbf {f}}_{k}^{\prime }(\varvec{ \theta }_{0})\right] ^{\top }{\mathop {\rightarrow }\limits ^{{\mathbb {P}}}} \sigma ^2 {\mathbf {I}}. \end{aligned}$$
Consequently, by the preceding conditions and making use of (A.14), we have, as \(n\rightarrow \infty\)
Now, as in [36], we may consider the assumption that
where \({\mathbf {V}}\) a positive definite and nonrandom matrix. Because
and by the martingale strong law [38], we infer that
So, the assumption (A.15) becomes simpler:
Similarly to [26] (Chapter 6), we have as \(n\rightarrow \infty\),
we then obtain, as \(n\rightarrow \infty\),
Thus the proof is complete. \(\square\)
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Papamichail, C., Bouzebda, S. & Limnios, N. Regression Analysis of Stochastic Fatigue Crack Growth Model in a Martingale Difference Framework. J Stat Theory Pract 14, 44 (2020). https://doi.org/10.1007/s42519-020-00110-x
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DOI: https://doi.org/10.1007/s42519-020-00110-x
Keywords
- Stochastic regression model
- Fatigue crack growth
- Least squares estimators
- Consistency
- Asymptotic normality
- Martingale differences
- Markov process
- Semi-Markov process