## 1 Introduction
### Burr-XII Distribution  
The Burr XII distribution is a flexible and versatile distribution that is useful for modeling a wide range of data types. Its parameters allow it to exhibit different kurtosis and tail properties to adapt to different shapes, and the distribution has non-monotone hazard rates. In addition, both the cumulative and reliability function of the Burr-XII distribution have closed-form expressions, making it easier to compute percentiles and the likelihood function based on censored data, which makes it a valuable tool in statistical modeling and analysis [14]. A random variable \(X\) is said to follow the Burr \((a,b)\) distribution if its probability density function (PDF) and cumulative distribution function (CDF) are given by the following expressions, respectively:  
\[f(t;a,b)=abt^{b-1}(1+t^{b})^{-(a+1)},t>0;a,b>0, \tag{1}\]  
\[F(t;a,b)=1-(1+t^{b})^{-a},t>0, \tag{2}\]  
where \(a,b\) are two positive shape parameters of the Burr-XII distribution. In addition, the reliability and hazard rate function take the form as:  
\[R(t;a,b)=(1+t^{b})^{-a},t>0, \tag{3}\]  
\[h(t;a,b)=\frac{abt^{b-1}}{1+t^{b}},t>0. \tag{4}\]  
In the density plots in Figure 1, it can be observed that when \(b\leq 1\), the density curve is strictly decreasing, while the Burr-XII distribution density curve is unimodal when \(b>1\) and at this time, a larger \(a\) corresponds to a higher peak height of the curve. In addition, Figure 2 shows the hazard rate function plots. As shown in the picture, when \(b\leq 1\), the hazard rate function decreases with \(t\) becoming larger, while the plot has a single peak when \(b>1\). The peak is higher when the value of parameter \(a\) becomes larger. Figure 1: PDF plots of Burr-XII distribution under various parameter configurations. Figure 2: Hazard rate function plots of Burr-XII distribution under various parameter configurations. Recently, owing to its versatility in applied contexts like reliability analysis, quality assurance, and quantum systems, the Burr-XII distribution has attracted significant interest from several scholars. There are many studies for Burr-XII distribution based on various life tests and censoring schemes. [PERSON] [15] derived both the MLE and Bayesian estimation approaches for the Burr-XII distribution parameters under progressive Type-II censoring scenarios. [PERSON] et al. [16] exploited the MLE and Bayes estimation within an improved adaptive Type-II progressive censoring framework under the Burr-XII distribution. [PERSON] and [PERSON] [17] investigated the Burr-XII distribution in competitive risk modeling under adaptive Type-II progressive censoring. Bayesian point estimates under various loss functions and the corresponding highest posterior density credible intervals were obtained through MCMC simulations. Although there are a large amount of statistical inferences related to the Burr-XII distribution along with censored data, they mainly focused on Type-II censoring or other progressive censoring schemes, and studies on the Burr-XII distribution under middle censoring are currently limited. [PERSON] [18] analyzed middle-censored data from the Burr-XII distribution. MLEs of two parameters were obtained, and Bayesian inferences under a Gamma prior were established through the application of [PERSON]'s approximation technique. However, it did not consider the failure reasons. Thus, based on that, in this paper we will introduce independent competing risks to study the middle-censored data under Burr-XII distribution. To our knowledge, no existing studies have comprehensively investigated the parameter-estimation challenges for Burr-XII distributions considering both middle-censored data and competing risks models. In this paper, we combine the competing risks with middle-censored data to address the parameter-estimation problems for the Burr-XII distribution. Compared to [17], we introduce a new censoring scheme and consider different numbers of failure causes. Different from [18], we introduce a competing risks model to the analysis framework.