Order Statistics Properties of the Two Parameter Lomax Distribution

In this paper we study the distribution of order statistics of the two parametric Lomax distribution. We consider the single and product moment of order statistics from Lomax distribution. Also, we establish some recurrence relation for single moments of order statistics. The exact analytical expressions of entropy, residual entropy and past residual entropy for order statistics of Lomax distribution is derived. AMS Subject Classification: 62G30; 94A17; 60E05.


Introduction
Order statistics have been used in wide range of problems, including robust statistical estimation and detection of outliers, characterization of probability distribution, goodness of fit-tests, quality control, analysis of censored sample, etc; see (Arnold et al., 1992;Beirlant et al., 1997;Tahir et al., 2015).The use of recurrence relations for the moments of order statistics is quite well known in statistical literature (see for example Arnold et al., (1992), Malik et al. (1998).For improved form of these results, Samuel and Thomes (2000), Arnold et al. (1992) have reviewed many recurrence relations and identities for the moments of order statistics arising from several specific continuous distributions such as normal, Cauchy, logistic, gamma and exponential.The Lomax distribution is often used in business, economics and actuarial modeling.It was used by Lomax (1954)  The following functional relationship exists between p.d.f and c.d.f of Lomax distribution: The idea of entropy of a random variable was developed by Claude Shannon (1948) for the first time in information theory.The differential entropy of a continuous random variable with density function ( ) is defined as Analytical expression for univerate distribution are discussed in references such as Laz and Rathie (1978), Nadarajah and Zagrafos (2003), etc.Also the information properties of order statistics have been studied by Wong and Chen (1990), Park (1995), Ebrahimi et al. (2004), etc.
In case on has information about the current age of the component, which can take into account for measuring its uncertainty, then the measure given in (1.5) is not suitable.A more realistic approach which make the use of the current age into account is described by Ebrahimi (1996) and is defined as where ̅ ( ) is the survival function.For (1.6) reduces to (1.5).
In many realistic situation uncertainty is not necessarily related to future but can also refer to past.Based on this idea, Crescenzo and Longobardi (2004) develop the concept of past entropy over ( ) If denote the lifetime of a component or of living organism, then the past entropy of is defined as where ( ) is the cumulative distribution function.For (1.7) reduces to (1.5).

Distribution of Order Statistics
Let be a random sample of size from the Lomax distribution and let denotes the corresponding order statistics.Then the pdf of is given by [see David and Nagaraja (1981)  Following two theorems gives the distribution of the order statistics from the Lomax distribution.
Theorem 2.1: Let ( ) and ( ) be the cdf and pdf of the Lomax distribution.Then the density function of the order statistics say ( ) is given by Proof: See Appendix A Theorem 2.2: Let and for be the and order statistics from the Lomax distribution.Then the joint pdf of and is given by Proof: See Appendix A

Single and Product Moments
In this section, we derive explicit expressions for both of the single and product moments of order statistics from the Lomax distribution.
Theorem 3.1: Let be a random sample of size from the Lomax distribution and let denote the corresponding order statistics.Then the moments of the order statistics for denoted by ( ) is given by where the value of is chosen in such a way that ./ Proof: See Appendix B Now we derive recurrence the relation for single moments.
Theorem 3.2: Let be a random sample of size from the Lomax distribution and let denote the corresponding order statistics.Then for , we have the following moment relation: ( Therefore the variance of the order statistic can be obtained easily by using the relation ) ) The mean, variance and other statistical measure of the extreme order statistics are always of great interest.Taking one can obtain the mean of smallest order statistics: Also, second order moment of the smallest order statistic can be obtained as: (Ghitany et al., 2007;Giles et al., 2013)us properties and applications of Lomax distribution one should refer to(Ghitany et al., 2007;Giles et al., 2013).