## 1 Introduction  
Rolling element bearings (REBs) are one of the most common mechanical components of rotary machines. The failure of these components is the main reason for 45% to 55% of machines breakdown [1]. This is why researchers have paid more attention to predicting the remaining useful life (RUL) of REBs over the past two decades. The RUL of REBs is defined as the remaining time until a vibration feature reaches the predetermined failure threshold (FT) [2], which is often assumed to be a constant value in the literature. Therefore, the accuracy of the RUL prediction depends on selecting a proper FT. Furthermore, the determination of FTs for various vibration features is a challenging issue. In general, to achieve protective or predictive goals in preventing the breakdown of rotary machines, the FT is determined experimentally, based on the manufacturers' recommendation or using the recommendations of standard codes (e.g., ISO 10816-3 [3]). The recommendations in ISO 10816-3 and other similar codes correspond to different fault types in machines in the frequency range of 10 to 1000 Hz. They are thus not specifically for the failure of REBs. Therefore, employing a suitable and proper FT for REBs leads to the more reliable and economical operation of rotary machines. Reviewing the literature, three different approaches can be considered for defining the FT: (1) as a constant value of a vibration feature, (2) as a probability distribution of a vibration feature or (3) as a specific level of physical damage. In most research studies on the RUL prediction of REBs, the FT is defined as a constant value of a vibration feature (e.g., the root mean square (RMS) or the time signal entropy) [4; 5; 6; 7; 8]. This approach is the most convenient method for this issue. For instance, in the PRONOSTIA dataset [9], which is a set of run-to-failure tests of REBs conducted by the FEMTO-ST institute in 2011, the constant value of the vibration peak (20 g) was employed as the stop criterion of the tests. After the publication of this dataset at the PHM 2012 conference, many researchers used it to validate their studies [10; 11; 12]. Most of them have used the test stop criterion as the FT in their RUL prediction problems. In the second approach, the FT is defined as a probability distribution of a vibration feature. This approach has been widely used in reliability literature. In this regard, [PERSON] and [PERSON] [13] studied the effect of different FT distributions on equipment reliability. [PERSON] et al. [14] used the gamma distribution of the degradation indicator for the FT, the parameters of which were determined by experts, to estimate the probability of chock valves' RUL accurately. In another study, [PERSON] and [PERSON] [15] studied the failure of an element in the crack propagation problem. They proposed a probability distribution of the crack size to define a failure threshold in the fatigue test results. Using a specific level of physical damage as a FT is less common than the previous approaches. Another dataset that has been employed in the RUL prediction literature is that of the REB run-to-failure tests conducted by the Center for Intelligent Maintenance Systems (IMS) at the University of Cincinnati [16]. In this dataset, the test stopping criterion was defined as a specific contaminant level in the REB lubrication feedback system. Reaching this level led to the activation of an electrical switch that stopped the test running. The dataset included three run-to-failure experiments on four similar parallel REBs mounted on a single shaft. These experiments stopped with the same physical damage criterion, which was explained earlier. However, investigating the results of the vibration records revealed a significant difference between the levels of various vibration features at the end of these three tests. This means that the stopping criterion in this dataset employed led to extremely different results from the first approach, which was explained earlier.