*Article* **Reliability Analysis Based on a Gamma-Gaussian Deconvolution Degradation Modeling with Measurement Error**

**Luis Alberto Rodríguez-Picón 1,\*, Luis Carlos Méndez-González 1, Roberto Romero-López 1, Iván J. C. Pérez-Olguín 1, Manuel Iván Rodríguez-Borbón <sup>1</sup> and Delia Julieta Valles-Rosales <sup>2</sup>**

> <sup>1</sup> Department of Industrial Engineering and Manufacturing, Institute of Engineering and Technology, Autonomous University of Ciudad Juárez, Ciudad Juárez 32310, Mexico; luis.mendez@uacj.mx (L.C.M.-G.); rromero@uacj.mx (R.R.-L.); ivan.perez@uacj.mx (I.J.C.P.-O.); ivan.rodriguez@uacj.mx (M.I.R.-B.)

<sup>2</sup> Department of Industrial Engineering, New Mexico State University, Las Cruces, NM 30001, USA; dvalles@nmsu.edu

**\*** Correspondence: luis.picon@uacj.mx

**Abstract:** In most degradation tests, the measuring processes is affected by several conditions that may cause variation in the observed measures. As the measuring process is inherent to the degradation testing, it is important to establish schemes that define a certain level of permissible measurement error such that a robust reliability estimation can be obtained. In this article, an approach to deal with measurement error in degradation processes is proposed, the method focuses on studying the effect of such error in the reliability assessment. This approach considers that the true degradation is a function of the observed degradation and the measurement error. As the true degradation is not directly observed it is proposed to obtain an estimate based on a deconvolution operation, which considers the subtraction of random variables such as the observed degradation and the measurement error. Given that the true degradation is free of measurement error, the first-passage time distribution will be different from the observed degradation. For the establishment of a control mechanism, these two distributions are compared using different indices, which account to describe the differences between the observed and true degradation. By defining critical levels of these indices, the reliability assessment may be obtained under a known level of measurement error. An illustrative example based on a fatigue-crack growth dataset is presented to illustrate the applicability of the proposed scheme, the reliability assessment is developed, and some important insights are provided.

**Keywords:** deconvolution; gamma process; lifetime; measurement system analysis; reliability estimation
