Characterizations through generalized and dual generalized order statistics, with an application to statistical prediction problem
Introduction
Kamps (1995) introduced the concept of generalized order statistics (GOS) as a unification of several models of ascendingly ordered random variables (RVs) with different interpretations. Ordinary order statistics (OOS), records (ordinary upper record values, when ), sequential order statistics (SOS), ordering via truncated distributions and censoring schemes can be discussed as they are special cases of the GOS. Ever since the advent of GOS, the use of such a model has been steadily growing along the years because it is a more capable of describing different reliability models.
The concept of dual generalized order statistics (DGOS) was introduced by Burkschat et al. (2003) to enable a common approach to descendingly ordered RVs like reversed OOS and lower records models.
In this work, we consider a wide subclasses of GOS and DGOS, known as -generalized order statistics (GOS) and dual generalized order statistics (DGOS), which contain many important models of ordered RVs such as OOS, records, SOS and type II censored OOS. Let be an arbitrary continuous distribution function (DF), with the probability density function (PDF) and survival function . Then the RVs () are said to be -GOS, if their joint probability density function (JPDF) is given by (cf. Kamps, 1995) where , (the left endpoint of the DF ), (the right endpoint of the DF ) and . On the other hand, the RVs are said to be -DGOS, if their JPDF is given by (cf. Burkschat et al., 2003) where .
From now onward, we omit of the notations -GOS and -DGOS to make them less cumbersome. In this paper, we consider only the case when (i.e., we exclude the case of record values). When the , the marginal DFs of the th GOS (cf. Kamps, 1995) and DGOS (cf. Burkschat et al., 2003) are given, respectively, by and where . Clearly, (1.2) is obtained from (1.1) just by replacing by . Moreover, in (1.1), (1.2) by convention, we will write and .
Section snippets
Characterization results based on GOS and DGOS
Characterization of a probability distribution is to find a unique property enjoyed by that distribution. Different results of characterization and its applications in terms of GOS and OOS are derived by many authors. Among these authors are Fagiuoli et al. (1999), Wesolowski and Ahsanullah (2004), Oncel et al. (2005), Arnold et al. (2008), Beutner and Kamps (2008), Samuel (2008), Castaño Martínez et al. (2012), Khan et al. (2012), Tavangar and Hashemi (2013) and Shah Imtiyaz et al., 2014,
An application to the prediction problem
One of the most important problems in statistics, is to predict future events based on past or current events. The prediction point and interval for ordered RVs are used extensively in reliability theory, survival studies and industrial applications for predicting the number of defective items to be produced during future production process. An excellent review of development on prediction problems till late 90 s can be found in Kaminsky and Nelson (1998). Point and interval prediction have
CRediT authorship contribution statement
Imtiyaz A. Shah: Conceptualization, Methodology, Writing - original draft. H.M. Barakat: Data curation, Investigation, Methodology, Software, Validation, Writing - review & editing. A.H. Khan: Writing - review & editing.
Acknowledgments
The authors wish to thank the Associate Editor and the two referees for the insightful review and helpful suggestions which led to this much improved version.
References (27)
Inference for Weibull distribution under generalized order statistics
Math. Comput. Simulation
(2010)Generalized order statistics from exponential distribution
J. Statist. Plann. Inference
(2000)- et al.
Exact prediction intervals for future exponential lifetime based on random generalized order statistics
Comput. Math. Appl.
(2011) - et al.
Prediction intervals
- et al.
Characterization through distributional properties of order statistics
J. Egyptian Math. Soc.
(2012) - et al.
Switching record and order statistics via random contraction
Statist. Probab. Lett.
(2005) The generalized order statistics from exponential distribution
J. Statist. Res.
(2006)- et al.
Statistical Prediction Analysis
(1975) - et al.
Bayes two-sample prediction of generalized order statistics with fixed and random sample size
J. Stat. Comput. Simul.
(2010) - et al.
Some characterizations involving uniform and powers of uniform random variables
Statistics
(2008)
Prediction for future exponential lifetime based on random number of generalized order statistics under a general set-up
Statist. Papers
Random contraction and random dilation of generalized order statistics
Comm. Statist. Theory Methods
Dual generalized order statistics
Metron
Cited by (5)
Predictive inference of dual generalized order statistics from the inverse Weibull distribution
2023, Statistical PapersIdentification of Wind Loads Through Train Statistical Analysis
2023, Lecture Notes in Networks and SystemsSome Characterizations of the Exponential Distribution by Generalized Order Statistics, with Applications to Statistical Prediction Problem
2022, Revista Colombiana de EstadisticaPredicting future order statistics with random sample size
2021, AIMS Mathematics