## 1 Introduction  
### Middle-Censored Data and Competing Risks  
Censoring occurs frequently in life test studies and cannot be completely avoided because of various practical constraints. Middle-censoring, a general censoring mechanism, was proposed by [PERSON] and [PERSON] [1]. In middle-censoring, the precise failure time of the tested component cannot be detected when it occurs within a random interval. That is, the exact lifetimes of some units are observable, while for others we only know the intervals. It is supposed that there are \(n\) identical units in a life test. The lifetimes of these units are defined as \(T_{1},T_{2},\ldots,T_{n}\), and the \(i\)-th corresponding random interval is \([L_{i},R_{i}]\). If \(T_{i}\in[L_{i},R_{i}]\), we cannot know the actual lifetime of the \(i\)-th unit, then \(\delta_{i}=0\). Otherwise, \(\delta_{i}=1\) denotes that the exact lifetime of the \(i\)-th unit can be observed directly. We can reorder the data, placing the complete data at the front, then the data can be described as:  
\[(T_{1},\delta_{1}),(T_{2},\delta_{2}),\ldots,(T_{n_{1}},\delta_{n_{1}}),([L_{ n_{1}+1},R_{n_{1}+1}],\delta_{n_{1}+1}),\ldots,([L_{n_{1}+n_{2}},R_{n_{1}+n_{2}}], \delta_{n_{1}+n_{2}}).\]  
[PERSON] and [PERSON] [1] not only put forward the middle-censoring scheme, but also developed an algorithm for obtaining the self-consistent estimator for the middle-censored data. The maximum likelihood estimator (MLE) in a non-parametricsetting was also obtained. In addition, [PERSON] and [PERSON] [2] developed an approximation to the distribution function of the middle-censored data. Then, they also suggested a straightforward alternative estimator for the lifetime distribution function, analyzed its performance under middle-censored data, and established its consistency and weak convergence. Moreover, several scholars have delved into middle-censored data within the parametric situation. [PERSON] et al. [3] considered a middle-censoring scheme in the case where the lifetime of the tested unit follows the exponential distribution. The maximum likelihood estimator was derived and its consistency and asymptotic properties were rigorously established. Additionally, the Bayes estimator was derived under a Gamma prior distribution. [PERSON] and [PERSON] [4] studied a discrete set-up with middle-censored data, where the survival time, the left endpoint of the censoring interval, and the length of the interval are all variables modeled by geometric distributions. They derived the MLE of the unknown parameter using the EM algorithm. Bayesian estimator and the corresponding credible intervals were also obtained. [PERSON] and [PERSON] [5] developed the estimation under the middle-censoring paradigm in a multinomial distribution with \(k\) distinct possible outcomes. [PERSON] et al. [6] concentrated on the Weibull and exponential distributions to explore the robustness of their parameter estimates and discovered that estimations for middle-censoring exhibited greater robustness when contrasted with those for right censoring. In practical applications, failure causes are typically pre-existing risk factors that may interact competitively. For example, in engineering reliability testing, environmental stressors such as temperature and humidity often compete to induce component failure. Statistical literature refers to this competitive failure mechanism as 'competing risks', emphasizing how multiple antecedent factors compete to be the first to exceed critical thresholds and cause system breakdown. To consider the different reasons for failure, many studies combine competing risks with middle-censoring. Based on middle-censored data with two independent competing risks, [PERSON] [7] explored various estimations of the parameter for exponential distribution. [PERSON] et al. [8] considered different estimations under middle-censored independent competing risks data with exponential distribution. Both the fix-point method and the EM algorithm were used to calculate the MLE.