Ordered Random Variable
A Ordered Random Variable is a Random Variable that is an Ordered Set in which each elements is an order statistic.
- Example(s):
- Counter-Example(s):
- See: Stochastic Ordering, Order Statistics, Common Order statistics, Sequential Order statistics.
References
2019
- (Nadarajah et al., 2019) ⇒ Saralees Nadarajah, Emmanuel Afuecheta and Stephen Chan (2019). Ordered random variables. OPSEARCH, 1-23. DOI:10.1007/s12597-019-00355-6
2011a
- (Wiper et al., 2011) ⇒ Michael Peter Wiper, Rosa Elvira Lillo Rodriguez, and Nuria Torrado Robles (2011). "On stochastic properties between some ordered random variables" DES - Working Papers. Statistics and Econometrics. WS ws110603, Universidad Carlos III de Madrid. Departamento de Estadística.
- QUOTE: Models of ordered random variables are widely used in statistical modelling and inference. If the random variables [math]\displaystyle{ X_1, \cdots, X_n }[/math] are arranged in ascending order of magnitude, then the i'th smallest of [math]\displaystyle{ X_i }[/math]'s is denoted by [math]\displaystyle{ X_{i:n} }[/math]. The ordered quantities[math]\displaystyle{ X_{1:n} \leq X_{2:n} \leq \cdots \leq X_{n:n} }[/math] \quad\quad (1.1)
are called order statistics (OS), and [math]\displaystyle{ X_{i:n} }[/math] is the i'th order statistic. These random variables are of great interest in many areas of statistics, in particular, there is a very interesting application of OS’s in reliability theory. The [math]\displaystyle{ (n−k+1) }[/math]’th OS in a sample of size n represents the life length of a k-out-of-n system which is an important technical structure. It consists of n components of the same kind with independent and identically distributed life lengths. All n components start working simultaneously, and the system works, if at least k components function; i.e. the system fails, if [math]\displaystyle{ (n − k + 1) }[/math] or more components fail. Special cases of k-out-of-n systems are series and parallel systems.
- QUOTE: Models of ordered random variables are widely used in statistical modelling and inference. If the random variables [math]\displaystyle{ X_1, \cdots, X_n }[/math] are arranged in ascending order of magnitude, then the i'th smallest of [math]\displaystyle{ X_i }[/math]'s is denoted by [math]\displaystyle{ X_{i:n} }[/math]. The ordered quantities
2011b
- (Bedbur, 2011) ⇒ Stefan Bedbur (2011). "Models of Ordered Random Variables and Exponential Families" (Doctoral dissertation, Hochschulbibliothek der Rheinisch-Westfälischen Technischen Hochschule Aachen).
- QUOTE: Models of ordered random variables play an important role in many statistical applications. In insurance mathematics, different models of record values are helpful to describe largest claims to an insurance company, whereas, in reliability theory, the models of common order statistics and sequential order statistics are of particular interest in modeling so called k-out-of-n systems.
2002
- (Gangnon & King, 2002) ⇒ Ronald E. Gangnon and William N. King (2002). "Minimum distance estimation of the distribution functions of stochastically ordered random variables". Journal of the Royal Statistical Society: Series C (Applied Statistics), 51(4), 485-492. DOI:10.1111/1467-9876.00282