New Characterizations of the Pareto Distribution

Characterization results have great importance in statistics and probability applications. Some characterizations of Pareto of the first kind and Pareto of the second kind distributions are presented by using conditional expectation in terms of their failure (hazard) rate. We also provide two characterization theorems based on the th truncated moments.


Introduction
In recent years order statistics and their moments have assumed considerable interest, the moments of order statistics have been tabulated quite extensively for several distributions, for example see Arnold et al. (1992) and David (1981).Many papers dealing with characterization through properties of order statistics are appeared, see for example Khan and Abouammoh (1999), Malik et al. (1988), Lin (1988), Kamps ((1991), (1995)), and Mohie El-Din et al. (1991).
Khan and Abu-Salih (1989) have characterized many well-known continuous probability distributions such as Pareto and power function distributions through conditional expectation of functions of order statistics.Ahsanullah and Raqab (2004) have characterized continuous distributions by conditional expectation of some functions of generalized order statistics.Ahsanullah and Hamedani (2007) characterized beta of the first kind and the power function distribution using order statistics and order statistics respectively.Hamedani et al. (2008) characterized certain univariate distributions using truncated moments ( ) .We like to mention here the works of Galambos and Kotz (1978), Kotz and Shanbag (1980), Ahsanullah (1989) statistics.Then the of ( ) , the joint of ( ) and ( ) and the conditional of ( ) given ( ) are, respectively, see Arnold et al. (1992).
In section 2, the Pareto of the first kind, Pareto of the second kind distributions are to be characterized through truncated moments of order statistics given by:

(Necessity):
Observe that Using equation ( 1.3), we obtain 2), we obtain Notice that equation (2.4) can be reduced to Differentiating the both sides of equation (2.5) with respect to , we or equivalently ( ) Integrating the both sides of equation (2.6) with respect to , we obtain From the fact that ∫ ( ) Then hence Which is the of the Pareto distribution of the first type.
This completes the proof.

Characterization of Pareto of the Second Kind Distribution
In the sequel, we shall use the following symbol ( ) The and the of the Pareto distribution of the second type are respectively,

. Characterization Theorems 2 . 1 1
Characterization of Pareto of the First Kind Distribution The and the survival function ( ) of the Pareto distribution of the first type are respectively, Let be a nonnegative continuous random variable with distribution function ( ), survival (reliability) function ( ), density function ( ) and Failure (hazard) rate function ( ).Let ( ) ( ) ( ) denote the order statistics of a random sample of size from ( ).The random variable has the Pareto distribution of the first type if and only if , Oncel et al. (2005) and Wesolowski and Ahsanullah (2004).Ahsanullah (2009) characterized several univariate distributions by the moments of the ( ) ( ).Let ( ) ( ) ( ) be the corresponding order

2.2
Letbe a nonnegative continuous random variable with distribution function ( ), survival (reliability) function ( ), density function ( ) and Failure (hazard) rate function ( ).Let ( ) ( ) ( ) denote the order statistics of a random sample of size from ( ).The random variable has the Pareto distribution of the second type if and only if