CHARACTERIZATIONS OF POWER DISTRIBUTIONS VIA MOMENTS OF ORDER STATISTICS AND RECORD VALUES

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1 APPLICATIOES MATHEMATICAE 26,41999, pp Z. GRUDZIEŃ and D. SZYALLublin CHARACTERIZATIOS OF POWER DISTRIBUTIOS VIA MOMETS OF ORDER STATISTICS AD RECORD VALUES Abstract. Power distributions can be characterized by equalities involving three moments of order statistics. Similar equalities involving three moments of -record values can also be used for such a characterization. The case of samples with random sizes is also considered. 1. Introduction. Too and Lin [8 have given a characterization of the uniform distribution by an equality involving only two moments of order statistics. We extend that result to power distributions. Moreover, we give a characterization of power distributions in terms of moments of -record values. In Sections 4 and 5 we treat the characterization problem when sample sizes are random cf. [1, [6, [9. 2. A characterization of power distributions. Let X :n be the th smallest order statistic of a random sample X 1,...,X n from a distribution F. Let m be a negative integer. We start with the problem of characterizing the power distribution function F defined as follows cf. [1: 2.1 Fx = 1 1+mx 1/m, x, 1/m. Theorem1. With the above notation suppose that EX:n 2 < for some pair,n. Then the equality 2.2 EX:n 2 2 [ n [ EX :n m EX :n m n m [ + 1 [ n [ m 2 2n [ +1 =, n 2m [ n m [ where n [ = nn 1...n +1, holds iff F is given by Mathematics Subject Classification: Primary 62E1, 6E99. Key words and phrases: sample, record values, power, uniform distributions. [467

2 468 Z. Grudzień and D. Szynal Proof. Let F 1 t = inf{x : Fx t, t,1}. Taing into account that 2.3 EX l :n = n! 1!n! F 1 t l t 1 1 t n dt, l 1, we see that E X :n <, and E X :n m <. Furthermore, when F is given by 2.1, we find that EX :s = s 2.4 [B,s m +1 m B,n +1, n s n m. and 2.5 EX:n 2 = n m 2 [B,n 2m+1 2B,n +1 +B,n +1, where Ba,b is the Beta function, and so 2.2 holds true. Conversely, assume that 2.2 holds. Applying 2.3 we see that 2.2 can be written as F 1 t 1 t m 1 m which implies that Fx is given by t 1 1 t n dt =, When m = 1 Theorem 1 reduces to the following characterization of the uniform distribution. Corollary1cf. [8. Let EX:n 2 < for some pair,n. Then 2.2 EX:n 2 2 n+1 EX +1 +1:n+1 + n+1n+2 = iff Fx = x on,1. In proving 2.2 we use the equality n EX :n +EX +1:n = nex :n 1 with = 1 cf. [2. Using 2.4 and 2.5 we obtain the following characterizing conditions. Theorem1. Under the assumptions of Theorem 1 the distribution function F is given by 2.1 iff EX :s = 1 [ s [ 1, s = n, n m, m s m [ EX:n 2 = 1 [ n [ n [ m n 2m [ n m [

3 Characterizations of power distributions 469 The same results can be derived for the distribution 2.6 Fx = 1 1+mx 1/m, x >, where m is a positive integer cf. [1. In this case 2.2 holds with n 2m >. 3. Characterizations in terms of moments of -record values. Let {X n, n 1} be a sequence of i.i.d. random variables with a common distribution function F. For a fixed integer 1 we define cf. [3 the sequence of -record values as follows: Y n = X L n:l n+ 1, n, where the sequence {L n, n 1} of -record times is given by L 1 = 1, L n+1 = min{j : j > L n, X j:j+ 1 > X L n:l n+ 1}, n. A characterization of F in 2.1 is contained in the following theorem. Theorem2. Let {X n, n 1} be a sequence of i.i.d. random variables with a common distribution function F such that E minx 1,...,X 2p < for a fixed 1 and some p > 1. Then F is given by 2.1 iff 3.1 EY n 2 2 m for n = 1,2, and [ m n + 1 m EY n n + Proof. Suppose that F is given by 2.1. Then we have EY n = n = 1 m 3.3 EY n 2 = n 1! n = 2m F 1 t[ log1 t n 1 1 t 1 dt [ m 1 m n n 1! = n n 1! = n n 1!m 2 = 1 m 2 [ 2m F 1 t 2 [ log1 t n 1 1 t 1 dt [1 t m 1 2 [ log1 t n 1 1 t 1 dt [ Γn 2m n 2Γn m n + 1 nγn n n 2 +1, m

4 47 Z. Grudzień and D. Szynal 3.4 n = n m 1 2m n 1 n which establishes 3.1. Conversely, assuming that 3.1 is satisfied we see that [ 2 F 1 t 1 t m 1 [ log1 t n 1 1 t 1 dt =. m Since the sequence { log1 t n, n 1}, is complete in L,1 cf. [7 we conclude that Fx is of the form 2.1. Theorem2. Under the assumptions of Theorem 2 the distribution function Fx is given by 2.1 for > 2m iff the following relations hold: s = 1 [ n s 1, s =, m, m s m for n = 1,2,... EY n 2 = 1 m 2 [ 2m n n 2 +1 Putting m = 1 we obtain the characterization results given in [6. For n = 1, m = 1 we obtain the result of Too and Lin [8. Similar considerations lead to the analogous characterizations for the distribution 2.6. amely, we have the following results. Theorem3. Let {X n, n 1} be a sequence of i.i.d. random variables with a common distribution function F such that E minx 1,...,X 2p < for a fixed 1 and some p > 1. Then Fx has the form 2.6 iff for 2m >, where m is a positive integer, 2 2 [ n 1 m m + 1 n n [1 2 m 2 + = 2m for n = 1,2,... Theorem3. Under the assumptions of Theorem 3, the distribution function F is given by 2.6 iff for 2m >, s = 1 [ n s 1, s =, m, m s m for n = 1,2,... EY n 2 = 1 m 2 [ 2m n n 2 +1,

5 Characterizations of power distributions 471 Letting m = 1 we obtain the following characterization result. Corollary2. Fx = 1 1+x 1, x >, iff [ n n n = Characterizations by moments of randomly indexed order statistics. LetX : betheth smallest orderstatistics of arandom sample X 1,...,X with common distribution function F, where is a random variable independent of {X n, n 1} with a probability function p = P[ =, = 1,2,... We write P = P[. In this section we give a characterization for the distribution 2.1 in terms of moments of order statistics with a random index. Theorem4. With the above notation, suppose that EX: 2 < for some and a given probability function p of. Then 4.1 EX: 2 2 [ E X : m m m EX : [ + 1 [E m 2 2m 2E [ m +1 =, [ where = , iff F is given by 2.1. Proof. Let F 1 t = inf{x : Fx t, t,1}. We have 4.2 EX l :n = n! 1!n! F 1 t l t 1 1 t n dt, l 1. Suppose that F is given by 2.1. Since is independent of {X n, n 1}, from 2.3 we have 4.3 EX : = 1 mp n= n! 1!n! [1 t m 1t 1 1 t n dtp[ = n = 1 m E m 1, [

6 472 Z. Grudzień and D. Szynal 4.4 E X : m m [ = 1 n m! mp 1!n m! and n= nn 1...n +1 n mn m 1...n m +1 1 t m 1t 1 1 t n m dtp[ = n = 1 [ E m 2m E [ m [ 4.5 EX: 2 = 1 [E m 2 2m 2E [ m +1. [ We see that 4.1 holds true. Conversely, assume that 4.1 holds. It can be written as 2 F 1 t 1 t m 1 t 1 1 t n dtp[ = n =, m n= which implies that F is given by 2.1. Using we have the following characterization conditions in terms of conditional moments of order statistics. Theorem4. Under the assumptions of Theorem 4 the distribution function F is given by 2.1 iff EX : = 1 [ E m m 1, [ E X : m m [ = 1 [ E m 2m E [ m [ and EX: 2 = 1 [E m 2 2m [ 2E m +1. [

7 Characterizations of power distributions 473 Corollary 3. Let be a random variable with probability function 4.7 P[ = n = αθn n, n = 1,2,...; α = 1 ln1 θ, θ,1. Then X has the distribution 2.1 iff m [θ m 1 α 1, EX 1: = 1 θ n m n n=1 E m X 1: m = 1 [ 2m θ 2m θ m θ 2m θ n m θ n α m n +θm α n n=1 n=1 EX1: 2 = 1 [ 2m m 2 θ 2m 2θ m αθ 2m θ n m n +2θm α n=1 n=1, θ n n +1. Remar. Putting m = 1 we obtain a characterization of the uniform distribution in terms of X 1:, which after using the equality EX1: 2 E +1 X 1: = αex EX 1: θ leads to the result of [9, i.e. EX1: [αex [ 3 EX 1: = α θ 2 1 θ 1 θ 1 2 ln1 θ. 5. Characterizations via moments of randomly indexed record statistics Theorem5. Let Y be the th record value, where is a positive integer-valued random variable independent of {X n, n 1}, and suppose that EY 2 <. Then F is given by 2.1 iff 5.1 EY 2 2 [ E Y EY m + 1 [1 2E m 2 +E =. 2m Proof. Suppose that F is given by 2.1. Since EY n l = n n 1! F 1 t l [ log1 t n 1 1 t 1 dt and and {X n, n 1} are independent, it follows that EY = 1 [ 1 m E, E Y = 1 m m [ E 2m E,

8 474 Z. Grudzień and D. Szynal n=1 EY 2 = 1 [E m 2 2E +1, 2m which establishes 5.1. Assuming now that 2.1 is satisfied we see that n 2 F 1 t 1 t m 1 n 1! m [ log1 t n 1 1 t 1 dtp[ = n =. Since the sequence { log1 t n, n 1} is complete in L,1 cf. [7 it follows that Fx has the form 2.1. Putting m = 1 we have the following characterization. Corollary4. Fx = x, x,1, iff EY 2 +2 E Y cf. [5 with m = 1. +E EY 2E = +1 Corollary 5. Let be a random variable with the probability function 4.7. Then X has the distribution 2.1 iff EY 2 2 [ E Y EY m + 1 [ m 2 1+2αlog 1 θ m αlog 1 θ 2m. 2m Remar. Fx = x, x,1, iff EY 2 +2 E Y αlog 1 θ+1 +1 EY αlog 1 θ+2 +2 =. Acnowledgements. The authors are grateful to the referee for useful comments which improved the presentation of the paper. References [1 M.Ahsanullah,Characteristicpropertiesoforderstatisticsbasedonrandomsample size from an exponential distribution, Statist. eerlandica , [2 H.A.David,OrderStatisticsinProbabilityandMathematicalStatistics,Wiley, 197.

9 Characterizations of power distributions 475 [3 W.DziubdzielaandB.Kopocińsi,Limitingpropertiesofthethrecordvalues, Zastos. Mat , [4 Z.GrudzieńandD.Szynal,Characterizationsofdistributionsbymomentsof order statistics when the sample size is random, Appl. Math.Warsaw , [5,, Characterization of continuous distributions in terms of moments of extremal statistics, J. Math. Sci , [6,, Characterizations of uniform and exponential distributions via moments of the th record values randomly indexed, Appl. Math.Warsaw , [7 G. D. L i n, Characterizations of distribution via moments of order statistics: A survey and comparison of methods, in: Statistical Data Analysis and Inference, Y. Dodge ed., orth-holland, 1989, [8 Y.H.TooandG.D.Lin,Characterizationsofuniformandexponentialdistributions, Statist. Probab. Lett , [9 G.D.LinandY.H.Too,Characterizationofdistributionsviamomentsoforder statistics when sample size is random, Southeast Asian Bull. Math , Department of Mathematics Maria Curie-S lodowsa University Pl. M. Curie-S lodowsiej Lublin, Poland szynal@golem.umcs.lublin.pl Received on ; revised version on