CHARACTERIZATIONS OF THE PARETO DISTRIBUTION BY CONDITIONAL EXPECTATIONS OF RECORD VALUES. Min-Young Lee. 1. Introduction

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1 Commun. Korean Math. Soc ), No. 1, pp CHARACTERIZATIONS OF THE PARETO DISTRIBUTION BY CONDITIONAL EXPECTATIONS OF RECORD VALUES Min-Young Lee Abstract. Let X 1, X 2, be a sequence of independent and identicall distributed random variables with continuous cumulative distribution function F x). X j is an upper record value of this sequence if X j > max{x 1, X 2,, X j 1 }. We define un) min{j j > un 1), X j > X un 1), n 2} with u1) 1. Then F x) 1 x θ, x > 1, θ < 1 if and onl if )E[X un+1) X um) ] θe[x un) X um) ], ) 2 E[X un+2) X um) ] θ 2 E[X un) X um) ], or ) 3 E[X un+3) X um) ] θ 3 E[X un) X um) ], n m Introduction Let {X n, n 1} be a sequence of independent and identicall distributedi.i.d) random variables with a common continuous distribution function F x) and probabilit densit function fx). Suppose Y n max{x 1, X 2,, X n } for n 1. We sa X j is an upper record value of this sequence if Y j > Y j 1, j > 1. B convention X 1 is an upper as well as a lower record value. We can transform from upper records to lower records b replacing the original sequence of random variables b { X j, j 1}. We define the record times un) b u1) 1 and un) min{j j > un 1), X j > X un 1), n 2}. The record times of the sequence {X n, n 1} are random variables and are the same as those for the sequence {F X n ); n 1}. We know that Received April 15, Mathematics Subject Classification: 62E15, 62E10. Ke words and phrases: absolutel continuous distribution, characterization, con -ditional expectation, pareto distribution, record value. The research was conducted b the research fund of Dankook universit in 2001.

2 128 Min-Young Lee the distribution of un) does not depend on F x). Hence, the distribution of un) can be determined b considering the uniform distribution F x) x. Also we will denote Ln) as the indices where the lower record value occur. We will call the random variable X PARθ 1, θ 2 ) if the corresponding probabilit cumulative function F x) of x is of the form ) θ2 θ1 1, θ 1 < x, θ 1 > 0, θ 2 > 0 F x; θ 1, θ 2 ) x 0, otherwise. Characterizations of the pareto distribution have been rarel studied in the literature. Nagaraja1977) characterized the pareto distribution that if E[hX L1 ) X L0 ] K) almost surel with respect to the distribution of X L0 where K) is a nondecreasing function on [c, d], then F x) is uniquel determined. In this paper we will give a characterization of the pareto distribution b considering conditional expectations of record values. 2. Results Theorem 1. F x) 1 x θ, x > 1, θ < 1 if and onl if 1) )E[X un+1) X um) ] θe[x un) X um) ], n m + 1. Theorem 2. F x) 1 x θ, x > 1, θ < 1 if and onl if 2) ) 2 E[X un+2) X um) ] θ 2 E[X un) X um) ], n m + 1. Theorem 3. F x) 1 x θ, x > 1, θ < 1 if and onl if 3) ) 3 E[X un+3) X um) ] θ 3 E[X un) X um) ], n m + 1.

3 Characterizations of the pareto distribution Proof Proof of Theorem 1. If F x) 1 x θ, then E[X un) X um) ] [see Ahsanullah1995)]. Hence 1) holds. Conversel, suppose 1) holds. From Ahsanullah formula 1995) we can obtain the following equation. 4) n m)! θ ln 1 F ) 1 F x) ln 1 F ) 1 F x) ) n m xfx)dx xfx)dx. Since F x) is absolutel continuous, we can differentiate n m + 1) times both sides of 4) with respect to and simplif, then we obtain the following equation. 5) f) θ1 F )) i.e. f) 1 F ) θ. Integrating both sides of 5) with respect to, we get F ) 1 θ. Proof of Theorem 2. If F x) 1 x θ, then E[X un) X um) ]. Hence 2) holds. Conversel, suppose 2) holds. From Ahsanullah formula we can obtain the following equation. 6) ) 2 n m + 1)! θ) 2 ln 1 F ) 1 F x) ln 1 F ) 1 F x) ) n m+1 xfx)dx xfx)dx. Since F x) is absolutel continuous, we can differentiate n m + 2) times both sides of 6) with respect to and simplif, then we obtain the following differential equation.

4 130 Min-Young Lee ) 2 f)) [ θ 2 1 F )) f) 21 F )) f 2 )) f )1 F )) 2 ] f 2 ) i.e. 7) 2)f) + 3θ 2 1 F )) + θ 2 f )1 F )) 2 f 2 ) 0. Therefore, there exists a unique solution of the differential equation 7) that satisfies the prescribed initial conditions F 1) 0, F 1) θ. B the existence and uniqueness Theorem, we get F ) 1 θ from 7). Proof of Theorem 3. If F x) 1 x θ, then E[X un) X um) ]. Hence 3) holds. Conversel, suppose 3) holds. From Ahsanullah formula we can obtain the following equation. 8) ) 3 n m + 2)! θ) 3 ln 1 F ) 1 F x) ln 1 F ) 1 F x) ) n m+2 xfx)dx xfx)dx. Since F x) is absolutel continuous, we can differentiate n m + 3) times both sides of 8) with respect to and simplif, then we obtain the following differential equation. ) 3 f)) θ 3 [ 61 F )) f 2 )) 3f )1 F )) 2 f 2 ) + 1 F )) f) f )1 F )) 3 f 6 ) + 31 F ))2 f))f))f 3 ) 3f 2 )f )) 2 1 F )) 3 ] f 6 )

5 Characterizations of the pareto distribution 131 i.e. 9) 3θ 2 + 3)f) + 7θ 3 1 F )) + θ 3 6f )1 F )) 2 θ 3 f )1 F )) 3 f 3 ) f 2 ) + θ 3 3f )) 2 1 F )) 3 f 4 ) 0. Therefore, there exists a unique solution of the differential equation 9) that satisfies the prescribed initial conditions F 1) 0, F 1) θ, F 1) θθ 1). B the existence and uniqueness Theorem, we get F ) 1 θ from 9). References [1] M. Ahsanuallah, Record values and the exponential distribution, Inn. Inst. Stat. Math ), no. A, [2], Record Statistics, Nova Science publishers, Inc, Commack NY. 1995). [3] H. N. Nagaraja, On a characterization based on record values, Austral. J. Statist ), Department of Mathematics Dankook Universit Cheonan , KOREA leem@dku.edu

CHARACTERIZATIONS OF THE PARETO DISTRIBUTION BY THE INDEPENDENCE OF RECORD VALUES. Se-Kyung Chang* 1. Introduction

JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 20, No., March 2007 CHARACTERIZATIONS OF THE PARETO DISTRIBUTION BY THE INDEPENDENCE OF RECORD VALUES Se-Kyung Chang* Abstract. In this paper, we