Characterization of the Linear Failure Rate Distribution by General Progressively Type-II Right Censored Order Statistics

In this article, we establish recurrence relations among single moments and among product moments of general progressively Type-II right censored order statistics and characterization for linear failure rate distribution using recurrence relations of single moments and product moments of general progressively Type-II right censored order statistics. Further, the results are specialized to the progressively Type-II right censored order statistics.


Introduction
In failure data analysis, it is common that some individuals cannot be observed for the full time to failure. the progressive Type-II right censored is a useful and more general scheme in which a specific fraction of individuals at risk may be removed from the study at each of several ordered failure times. Progressively censored samples have been considered, among others, by Balakrishnan et al. [6] and Davis and Feldstein [8]. Bain [5] derived analysis for the Linear Failure-Rate life-testing distribution. Aggarwala and Balakrishnan [3] derived recurrence relations for single and product moments of progressive Type-II right censored order statistics from exponential, Pareto and power function distributions and their truncated forms. Abd El-Aty and Marwa Mohie El-Din [1] derived recurrence relations for single and double moments of GOS from the inverted linear exponential distribution and any continuous function. Mokhlis et al. [13] derived recurrence relations for moments of GOS from Marshall-Olkin-Extended burr XII distribution. Mohie El-Din, and Kotb [12] derived recurrence relations for single and product moments of generalized order statistics for modified burr XII-Geometric distribution and characterization. Mohie El-Din et al. [11] derived Statistical Inference and Characterizations from Independent and Identical Exponential-Bernoulli Mixture Distribution. Athar et al. [4] discussed some new moments of progressively Type-II right censored order statistics from Lindley distribution. Saran and Pushkarna [9] derived moments of progressive Type-II right censored order statistics from a general class of doubly truncated continuous distributions. Abd El-Hamid et al. [2] derived inference and optimal design based on step-partially accelerated life tests for the generalized Pareto distribution under progressive Type-I censoring.
This scheme of censoring was generalized by Balakrishnan and Sandhu [7] The joint probability density function of the general  progressively Type-II right censored order statistics failure  times : :,% , &: :,% , … , : :,% , is given by ! ' ()*:+:,, ,…,' +:+:,, -, … , -= where, In this paper, we shall introduce recurrence relations among single and product moments of general progressively Type-II right censored order statistics. Characterization for linear failure rate distribution using recurrence relations of single and product moments of general progressively Type-II right censored order statistics. Let : :% 4 ()* ,4 (); ,…,4 + < &: :% 4 ()* ,4 (); ,…,4 + < ⋯ < : :% 4 ()* ,4 (); ,…,4 + be the ordered observed failure times in a sample of size − under general progressively Type-II right censored order statistics from the linear failure rate distribution with probability density function (pdf) is given by The corresponding cumulative distribution function (cdf) is given by It may be noted that from (2) and (3) the relation between pdf and cdf is given by, For any continuous distribution, we shall denote the single moment of the general progressively Type-II right censored order statistics in view of Eq.

Recurrence Relations for Single and Product Moments
In this section we introduce the recurrence relation for single and product moments of general progressively Type-II right censored order statistics from linear failure rate distribution.
In the next theorem we introduce the recurrence relation for single moment of general progressively Type-II right censored order statistics. where Now, integrating by parts gives Substituting Eq. (10) in Eq. (8) and simplifying, yields Eq.
This completes the proof.

Special case
Theorem (2.1) will be valid for the progressively Type-II right censored order statistics as a special case from the general progressively Type-II right censored order statistics when = 0, Substituting by Eq. (10) in Eq. (12) and simplifying, yields Eq. (11).
This completes the proof.

Special case
Theorem (2.2) will be valid for the progressively Type-II right censored order statistics as a special case from the general progressively Type-II right censored order statistics when = 0, where Now, integrating by parts we obtain Substituting by Eq. ( 16 ) in Eq. ( 14 ) and simplifying, yields Eq. (13).
This completes the proof.

Special case
Theorem (2.2) will be valid for the progressively Type-II right censored order statistics as a special case from the general progressively Type-II right censored order statistics when = 0,

Characterization for Single and Product Moments
In this section, we introduce the characterization of the linear failure rate distribution using the relation between pdf and cdf and using recurrence relation for single and product moments of general progressively Type-II right censored order statistics from linear failure rate distribution.
In the next theorem, we introduce the characterization of the linear failure rate distribution using relation between pdf and cdf. That is the distribution function of linear failure rate distribution.
This completes the proof.
In the next theorem, we introduce the characterization of the linear failure rate distribution using recurrence relation for single moment of general progressively Type-II right censored order statistics has introduced in the following theorems. where Integrating by parts, we obtain We get This completes the proof.
In the next two theorems, we introduce the characterize the linear failure rate distribution using recurrence relation for product moment of general progressively Type-II right censored order statistics.    This completes the proof.

Conclusion
We derived recurrence relations among single and product moments of general progressively Type-II right censored order statistics (Theorem 2.1, 2.2 and 2.3). Characterization for the random variable X following the linear failure rate distribution is obtained using the previous recurrence relations (Theorems 3.2, 3.3 and 3.4) and distribution function (Theorem 3.1). For future work, estimation for the scale and location parameters could be obtained by applying the best linear unbiased estimation to the previous results.