# The Combined-Unified Hybrid Censored Samples from Pareto Distribution: Estimation and Properties
## 1 Introduction  
Pareto distribution was introduced by [PERSON] [1] for the distribution of income. The importance of Pareto distribution lies in its applications in economics and reliability studies. [PERSON] [2] has given a wide historical account of Pareto distribution and its applications. Estimation and characteristics of Pareto distribution were investigated by many authors, among researchers, see for examples, [PERSON] [3], [PERSON] and [PERSON] [4], [PERSON], [PERSON] and [PERSON] [5], [PERSON], [PERSON] and [PERSON] [6], [PERSON] and [PERSON] [7], and [PERSON] [8]. [PERSON] et al. [9] have proposed a goodness-of-fit test for Pareto distribution when the observations are Type-II right censoring. [PERSON] [10] has constructed an interval estimation for Pareto distribution using a doubly Type-II censored sample. Recently, [PERSON] [11] has investigated the expected Bayesian estimation and its expected mean square error of Pareto distribution parameter under different loss functions and [PERSON] [12] has investigated the truncated, censored, and actuarial payment-type moments of the robust fitting of a single parameter Pareto distribution. A random variable \(X\) follows Pareto distribution \(P(k,\alpha)\) if its probability density function (pdf) is given by  
\[f(x)=ak^{\alpha}x^{-\alpha-1},\ \ \alpha>0,k>0,\ x\geq k, \tag{1}\]  
with the corresponding cumulative distribution function (cdf) is given by  
\[F(x)=1-\left(\frac{k}{x}\right)^{\alpha},\ \ x\geq k. \tag{2}\]  
The reliability function \(R(t)\) and hazard function \(H(t)\) are given, respectively, as  
\[R(t)=\left(\frac{k}{t}\right)^{\alpha}\text{ and }H(t)=\frac{\alpha}{t}. \tag{3}\]Quite a few techniques exist to estimate the shape parameter of Pareto distribution. Both of Type-I and Type-II censored are extensively used in practice. Type-I and Type-II censoring schemes can be merged to get the hybrid censoring scheme which was first introduced by [PERSON] [13]. The hybrid censoring scheme becomes quite important in the reliability and life testing problems, see [PERSON] et al. [14], [PERSON] and [PERSON] [15], [PERSON] and [PERSON] [16], [PERSON] et al. [17], [PERSON] et al. [18], and [PERSON] and [PERSON] [19]. [PERSON] and [PERSON] [20] have discussed theinerences based on Type-I and Type-II hybrid censoring schemes. Next, they have discussed some details on the generalized hybrid censoring and unified hybrid censoring schemes. Additionally, they have shown the adaption of the hybrid censoring schemes in competing risks set-up and in step-stress modeling. [PERSON] and [PERSON] [21] have discussed the parameter estimation from half-logistic distribution by using multiply Type-II hybrid censoring. [PERSON] and [PERSON] [22] have investigated the reliability characteristics of bathtub-shaped distribution under adaptive Type-I progressive hybrid censoring. [PERSON], [PERSON], and [PERSON] [23] have discussion the joint Type-I generalized hybrid censoring for estimating the two Weibull distributions. [PERSON], et al. [24] have distressed the estimation and prediction for Pareto distribution under Type-II progressive hybrid censoring scheme, while [PERSON] [25] has drawn inference based on Type-II hybrid censored data from a Pareto distribution. [PERSON] and [PERSON] [26] have suggested a combined hybrid censoring sampling scheme (CHCS) as follows: Assume \(n\) experimental units are placed under a certain experiment and let \(X_{m:n}\) and \(X_{l:n}\) denote the failure time of the \(m\)th and \(l\)th units, respectively, such that \((m,l)\in\{1,2,\ldots,n\}\), \((t_{1},\)\(t_{2})\in(0,\infty)\), \(m<l\), \(t_{1}<t_{2}\) and let \(t\) denote the termination time of the experiment. If the \(m\)th failure occurs before time \(t_{1}\), the experiment terminates at \(\min\{X_{l:n},t_{1}\}\), when the \(m\)th failure occurs in the interval \((\)\(t_{1}\), \(t_{2})\), then the experiment is stopped at \(X_{m:n}\) and finally when the \(m\)th failure occurs after time \(t_{2}\), the experiment is Stooped at \(t_{2}\).