## 1 Introduction  
Generally, the observed degradation of a performance characteristic of interest is an additive function of the true degradation, and some measurement error [1, 2, 3]. This means that in most cases, it is difficult to measure the degradation process over time due to imperfect measurement devices and environmental conditions. If the measurement system accuracy can be attained during the measuring process, then the general reliability assessment of the product under study may be deemed as precise. Nevertheless, in the presence of measurement error, the estimation and reliability assessment must consider the measurement error in the modeling such that the obtained conclusions may not be underestimated. Several models proposed in the literature consider the problem of obtaining the true degradation in the presence of measurement error with the common assumption is that the measurement error is independent of the degradation measurement [2, 4, 5]; given that the error comes from a measuring device that is independent of the true degradation. However, in some cases, it is considered that the measurement error is dependent on the true degradation [6, 7, 8, 9]. In addition, a common assumption is that the measurement error is normally distributed with mean zero and standard deviation \(\sigma\)[5, 10, 11]. The true degradation can be obtained by considering the joint distribution of the probability density function (PDF) of the observed degradation and the PDF of the measurement error as described in the works of [PERSON] [8], [PERSON] et al. [7], [PERSON] et al. [12], [PERSON] [PERSON] [13]. In these cases, the joint distribution is obtained either via joint conditional distributions or by the convolution of the observed degradation and the error [9]. In terms of stochastic modeling, the Wiener process is the most used in the literature to deal with measurement error. [PERSON] et al. [14] proposed a Wiener process model with logistic distributed measurement errors, the estimation of parameters was carried out with the Monte Carlo Expectation-Maximization method to estimate the related parameters. [PERSON] et al. [15] proposed a change-point Wiener degradation model with normally distributed measurement errors, they considered a Bayesian approach to estimate the parameters of interest. [PERSON] et al. [16] studied a Wiener degradation model with three sources of uncertainty, one being the measurement error, which is considered to be normally distributed. [PERSON] et al. [17] proposed a nonlinear Wiener process model with measurement error to estimate the remaining useful life of a cutting tool. The estimation of parameters of this model is extended by [PERSON] et al. [18]. [PERSON] and [PERSON] [19] also considered the Wiener process with measurement error but based on evidential variables. [PERSON] et al. [20] proposed a Wiener process model with normally distributed measurement errors and multiple accelerating variables. Models based on the inverse Gaussian process with measurement error have also been proposed. [PERSON] et al. [21], [PERSON] et al. [22] and [PERSON] et al. [23] studied the inverse Gaussian process with random effects and measurement errors to obtain lifetime estimations. A similar method for the inverse Gaussian process was also considered by [24] but under accelerating conditions. [PERSON] et al. [25] proposed a nonlinear adaptive inverse Gaussian process with measurement error to estimate remaining useful life. Another important modeling approach considers the deconvolution, which consists of the inverse process of the convolution in order to obtain an unknown PDF from two known PDFs. In such a case, the true degradation can be obtained by deconvoluting the PDF of the observed degradation and the known PDF of the measurement error. Although, the deconvolution has been used in different scientific disciplines leading to important applications such as in illumina BeadArrays [12,26], optical distortion [27] and image processing [28,29].