Bhatt Milind. B. Department of Statistics Sardar Patel University Vallabhvidyanagar Dist. Anand, Gujarat INDIA

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1 Journal of Statistical Sciee and Application, August 2016, Vol. 4, No , doi: / X/ D DAVID PUBLISHING Characterization of Pareto Distribution Through Expectation of Fution of Order Statistics Bhatt Milind. B Department of Statistics Sardar Patel University Vallabhvidyanagar Dist. Anand, Gujarat INDIA bhattmilind_b@yahoo.com For characterization of Pareto distribution one needs any arbitrary non constant fution only by approach of identity of distribution and equality of expectation of fution of random variable in place of approaches such as relation (linear) in (economic variation) reported and true iome, independey of suitable fution of order statistics, mean and the extreme observation of the sample etc. Examples are given for illustrative purpose. Keyword : Characterization; Pareto distribution Introduction Certain skew pattern appear in socioeconomic quantities such stock price fluctuation, personal iome, economic variation in reported iome and under-reporting error [See. Krishnaji (1970), Nagesh (1974)] have certain invariant properties for which Pareto distribution foun most duitable. Amongst many other Pareto distribution used to study skew pattern. Pareto distribution also used to study empiric phenomena such as occurree of natural resources, error clustering in communication circuit, size of firm, city, population and reliability theory. Independee of suitable fution of order statistics was used for characterization of Pareto distribution by Henrick (1970), Ahsanullah (1973, 1974 ), Shah (1981) and Dimaki (1993) where as Srivastava (1976) used mean and the extreme observation of the sample. Other attempts were made for characterization of exponential and related distributions assuming linear relation of conditional expectation by Beg (1974) and Dallas (1976), characterization of some types of distributions using recurree relations between expectations of fution of order statistics by Alli (1998) and characterization results on exponential and related distributions by Tavangar (2010) iluded characterization of Pareto distribution. This research note provides the characterization based on identity of distribution and equality of expectation of fution of order statistics for Pareto distribution with the probability density fution (p.d.f.). Corresponding author: Bhatt Milind. B, Department of Statistics, Sardar Patel University, Vallabhvidyanagar , Dist. Anand, Gujarat, INDIA. bhattmilind_b@yahoo.com.

2 Characterization of Pareto Distribution Through Expectation of Fution of Order Statistics 197 This work was supported by UGC Major Research Project : F.No.42-39/2013(SR) dated This research note provides characterization of Pareto distribution with probability density fution (pdf) ; ; 0,, (1.1) 0 ;, where are known constants and 1 is positive absolutely continuous fution and is everywhere differentiable fution is characterized. Note that is iome coentration use as measure of inequalities in iome distribution and is minimum level of iome. The aim of the present research note is to give path braking new characterization for Pareto distribution defined in (1.1) through expectation of fution of order statistics, using identity and equality of expectation. Characterization theorem provrd in section 2 with method for characterization as remark and section 3 devoted to applications for illustrative purpose. Characterization Theorem Let,,, be a random sample of size from distribution fution and let : : : be the set of corresponding order statistics. Assume that is continuous on the interval, where. Let : and : be be two distit differentiable and intregrable futions of first order statistic; : on the interval, where. and moreover $ : be non-constant fution of :. Then : : : (2.1) : is the necessary and sufficient condition for pdf of to be, defined in (1.1). Proof. Given, defined in (1.1), for necessity of (2.1) if : is such that : where is differentiable fution then using : ; pdf of first order statistic; : one gets gθ x : fx :,θdx : (2.2) Differentiating (2.2) with respect to θ on both sides and replacing : for and simplifying one gets x : : : : (2.3) : which establishes necessity of (2.1). Conversely given (2.1), let : ;θ be the pdf of first order statistic; : such that gθ x : kx :,θdx : (2.4) Sie : ; 1 is ireasing integrable and differentiable fution on the interval, with 0 the following identity holds : : : :. (2.5) Differentiating integrand : : with respect to : and simplifying after taking : :

3 198 Characterization of Pareto Distribution Through Expectation of Fution of Order Statistics as one factor one gets (2.5) as : and substituting derivative of : : : : in (2.6) one gets : : : : : (2.6) x : : (2.7) : where x : is as derived in (2.3). By uniqueness theorem from (2.4) and (2.7) kx :,θ (2.8) : Sie : ; 1 is ireasing integrable and differentiable fution on the interval, with 0 and sie : is ireasing fution for with 0. is satisfy only when range of X : is truated by θ from left and integrating (2.8) on the interval θ, on both sides, one gets 1 kx :,θdx : For 1, x :, reduces to, defined in (1.1). Hee sufficiey of (2.1) is established. Remark 2.1. Using given in (2.2) one can determine, by and pdf is given by X : X: X : X : X : (2.9) f, θ : :, aθ (2.10) where is decreasing fution for with 0 such that it satisfies M : log :. (2.11) : Remark 2.1. The theorem 2.1 for fution of first order statistics with remark 2.1also holds for random variable when Examples. Using method describe in remark 2.1 Pareto distribution through expectation of non-constant fution of order statistics is characterized as illustrative example and significant of unified approach of characterization result. Example 3.1 Characterization of Pareto distribution through the Minimum Variae Unbiased (UMVU) estimator of is given. Using (2.3) one gets : 1 : g :

4 Characterization of Pareto Distribution Through Expectation of Fution of Order Statistics 199 x : : : : : 1 2 : : and (2.9) of remark 2.1 with then and X : X X : : X : X : : log 1 X : : : T : 1 : : : f, θ : :, aθ Example 3.2 Characterization of Pareto distribution through the uniformly minimum variae unbiased (UMVU) estimator and maximum likelihood estimator (MLE) of such as ; mean, ; r th moment,, ; p th quantile, ; distribution fution and ; reliability fution is given. For the (UMVU) estimator c :; 1 c r 1 : ; 2 : 1 : ; 3 : 1 : and MLE : ; 5 ; 4 1 : 1 1 ; 6 n : 1 1 ; 7

5 200 Characterization of Pareto Distribution Through Expectation of Fution of Order Statistics and One gets c 1 :; 1 c r : ; 2 :; 3 :; 4 : 1 ; 5 1 : ; 6 : ; c 1 :; 1 c r 1 : ; 2 : : 1 : 1 ; 3 : : x : 1 : 1 ; 4 : ; 5 1 n 1 1 n : ; 6 1 n : 1 1 ; 7

6 Characterization of Pareto Distribution Through Expectation of Fution of Order Statistics 201 x : ; 1 c r : ; 2 : 1p c 1 : : : ; 3 : : ; 4 1 n X : t ; 5 ; 6 1 n X : t ; 7 respectively. Then by defining X : given in (2.9) and substituting T : as appeared in (2.11) for (2.10), f, θ is characterized. Example 3.3 In context of remark 2.2 characterization of Pareto distribution through p th quantile ; is given. Therefore 1 1 and from (2.3) and (2.9) of remark 2.1 with then and X c X 1 1 X X X X c : log 1 X : T 1

7 202 Characterization of Pareto Distribution Through Expectation of Fution of Order Statistics f, θ X X, aθ. Example 3.4 Using remark 2.2 the pdf f, θdefined in (1.1) can be characterized through non constant futions of θ such as ; 1 c 1 c r ; 2 ; 3 ; 4 1 ; 5 1 ; 6 ; 7 by using x : ; 1 1 ; 2 ; 3 ; 4 X C 1p ; 5 ; 6 ; 7 Then by defining X : given in (2.9) and substituting T : as appeared in (2.11) for (2.10), f, θ is characterized.

8 Characterization of Pareto Distribution Through Expectation of Fution of Order Statistics 203 Referee Ahsanullah, (1973). A characterization of the Pareto distribution. Canadian Journal of Statistics, Volume 1, Issue 1-2, pages , 1973 Ahsanullah (1974). Characterization of the Pareto distribution. communication in statistics, 3(10), , Ali, M. A. and Khan, A. H. (1998). Characterization of Some Types of Distributions, Information and Management Sciees, Vol. 9, No. 2, June, Beg, M. L and Kirmani, S. N. U. A, (1974). On a characterization of exponential and related distributions, Austral. J. Statist., 16 (3), 1974, Dallas, A. C. (1976). Characterization Pareto and power distribution, Ann. Ins. Statist. Math, 28, (1976), Part A, pp Dimaki,C.and Evdokia Xekalaki. (1993). Characterizations of the Pareto distribution based on order statistics. Stability Problems for Stochastic Models, Lecture Notes in Mathematics Volume 1546, pp Henrick John Malik, (1970). A characterization of the Pareto distribution. Scandinavian Actuarial Journal, Volume 1970, Issue 3-4, Krishnaji, N, (1970). Characterization of the Pareto Distribution Through a Model of Underreported Iomes. Vol. 38, No. 2, Mar., , Nagesh S. R., Michael J. H. and Marcello P. (1974). A Characterization of the Pareto Distribution. Institute of Mathematical Statistics Vol. 2, No.3, Page 599 of , Shah, S. M. and Kabe, D. G, (1981). Characterizations of Exponential, Pareto, Power Fution, BURR and Logistic Distributions by Order Statistics. Biometrical Journal, volume 23, Issue 2, pages Srivastava, M. S. (1965). Characterizations of Pareto s Distributions and 1. Ann. Math. Statist.,36, Tavangar, M. and Asadi, M. (2019). some new characterization results on exponential and related distributions, Bulletin of the Iranian Mathematical Society Vol. 36 No. 1 (2010), pp