International Journal of Mathematics And its Applications Volume 3, Issue 3–A (2015), 13–22. ISSN: 2347-1557 Available Online: http://ijmaa.in/ A p p l i c a t i o n s • I S S N : 2 3 4 7 - 1 5 5 7 • I n t e r n a t i o n a l J o u r n a l o f M a t h e m a t i c s A n d i t s International Journal of Mathematics And its Applications Characterization of the Generalized Weibull-Gompertz Distribution Based on the Upper Record Values Research Article A.H.El-Bassiouny 1 , Medhat EL-Damcese 2 , Abdelfattah Mustafa 1 and M.S.Eliwa 1* 1 Mathematics Department, Faculty of Science, Mansoura University, Egypt. 2 Mathematics Department, Faculty of Science, Tanta University, Egypt. Abstract: This paper introduces characterization of the generalized Weibull-Gompertz distribution based on the upper record values. Several properties are studied in this paper such as reversed (hazard) function, moments, maximum likelihood estimation, mean residual (past) lifetime. A real data set is analyzed. Keywords: Upper record values, reversed (hazard) function, generalized Weibull-Gompertz distribution, mean residual (past) lifetime. c  JS Publication. 1. Introduction Record values and the associated statistics are of importance in many life fields such that weather, economic, sports data “Olympic records or world records in sport”and life testing. Also, one of the most important applications of records in reliability studies, the structure k - out - of - n : G(F ) where the structure functions (fails) when or more out of the components are functioning (failing). This structure includes the cases of series and parallel systems when k = n and k = 1, respectively. Suppose we are interested in predicting the lifetime of the system when r components have failed, for some 1 ≤ r<k ≤ n. The lifetime of the system corresponds to the (n - k - r + 1) th future record value. On the other hand, in industry many products fail under stress, an electronic component ceases to function in an environment of too high temperature and a battery dies under the stress of time. But the precise breaking stress or failure point varies even among identical items. Hence, in such experiments, measurements may be made sequentially and only the record values are observed. Thus, the number of measurements made is considerably smaller than the complete sample size. This “measurement saving”can be important when the measurements of these experiments are costly if the entire sample was destroyed. So; many authors have studied record values and the associated statistics; see, for example, Balakrishnan and Chan [4], Pawlas and Szynal [8], Raqab [9], Al Zaid and Ahsanullah [2], Ahsanullah [1], Soliman et al. [10] and Grine Azedine [5]. ∗ E-mail: msabereliwa@mans.edu.eg