Let $Y_{1},\cdots,Y_{n}$ be order statistics from a distribution with parameter $\theta$. Assume that some of middle order statistics are lost, that is we only observed the data set $\textbf{Y}=\{Y_{1},\cdots,Y_{r},Y_{s},\cdots,Y_{n}\}$. We are going to reconstruct the $l(r < l < s)$th order statistic based on the data set $\textbf{Y}$. In this work, we propose two methods: Maximum Likelihood Reconstruction(MLR) and Bayesian Reconstruction(BR). To compute MLR, we obtain closed form for reconstruction $Y_{l}$. The next method for reconstruction $Y_{l}$ is the Bayesian approach. In this approach, we assume that the unknown parameter $\theta$ is viewed as the realization of a random variable which has a prior distribution. A numerical method has to be applied to compute the BR. Also, numerical example and Monte Carlo simulation study of the L\'{e}vy distribution are given to illustrate all the reconstruction methods discussed in this work. To evaluate the estimators, we compute Mean Square Reconstruction Errors for the MLR and BR reconstructions. Finally, we conclude BR is better than MLR.
