Characterization of probability distributions by conditional expectation of function of record statistics

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ORIGINAL ARTICLE

Characterization of probability distributions by

conditional expectation of function of record statistics

Zubdah-e-Noor, Haseeb Athar

*

Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh 202 002, India

Received 11 December 2012; revised 3 July 2013; accepted 7 July 2013 Available online 17 September 2013

KEYWORDS Record values;

Conditional expectation; Characterization; Probability distributions

Abstract In this paper, two general classes of distributions have been characterized through con-ditional expectation of power of difference of two record statistics. Further, some particular cases and examples are also discussed.

2010 MATHEMATICS SUBJECT CLASSIFICATION: 62E10; 62G30; 60E05

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1. Introduction

Record values are found in many situations of daily life as well as in many statistical applications. Often, we are interested in observing new records and in recording them, e.g. Olympic re-cords or world rere-cords in sports.

Record values are deﬁned by Chandler[1] as a model of successive extreme sequence of identically and independent random variables. It may also be helpful as a model for succes-sively largest insurance claims in non-life insurance, for highest water-levels or highest temperatures. Record values are also useful in reliability theory. To be precise, record values are de-ﬁned by means of record times. That is, those times have to be described at which successively largest values appear. Chandler [1] shows several properties of record value and notes their

Markovian structure. For detailed survey on the upper record values, one may refer to Arnold et al.[2]and Ahsanullah[3]. Suppose that X1, X2, . . . is a sequence of independent and

iden-tically distributed (i.i.d.) random variables with distribution func-tion (df) F(x) and probability density funcfunc-tion (pdf) f(x). Let Yn= max(min){X1, X2, . . ., Xn} for n P 1. We say Xjis an upper

(lower) record value of {Xn, n P 1}, if Yj> (<)Yj1, j > 1. By

deﬁnition, X1is an upper as well as lower record value. One can

transform the upper records to lower records by replacing the ori-ginal sequence of {Xj} by { Xj, j P 1} or (if P(Xj> 0) = 1 for

all j) by {1/Xj, j P 1}; the lower record values of this sequence will

correspond to the upper record values of the original sequence. The indices at which the upper record values occur are called the record times {U(n)}, n > 0, where U(n) = min{jŒj > U(n 1), Xj> XU(n1), n > 1} and U(1) = 1. The record times of

the sequence {Xn, n P 1} are the same as those for the sequence

{F(Xn), n P 1}.

The joint pdf of the r record values XU(1), XU(2), . . ., XU(r)is

given by fðx1; x2; . . . ; xrÞ ¼ fðxrÞ Yr1 i¼1 fðxiÞ FðxiÞ ; ð1:1Þ

* Corresponding author. Tel.: +91 5712701251. E-mail address:haseebathar@hotmail.com(H. Athar).

Peer review under responsibility of Egyptian Mathematical Society.

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for 1 < x1< x2< < xr1<1 and FðxÞ ¼ 1 FðxÞ.

The probability density function of XU(r)is given by

fUðrÞðxÞ ¼

1

CðrÞ½ lnðFðxÞÞ

r1

fðxÞ; a < x < b; ð1:2Þ and the joint pdf of XU(r)and XU(s)is given by

fUðrÞ;UðsÞðx; yÞ ¼ 1 CðrÞCðs rÞ½ lnðFðxÞÞ r1 ½ lnðFðyÞÞ þ lnðFðxÞÞsr1 fðxÞ FðxÞfðyÞ; a < x < y < b: ð1:3Þ

The conditional pdf of XU(s)given XU(r)= x, 1 6 r < s is[4]

fsjrðyjxÞ ¼ 1 Cðs rÞ½ ln FðyÞ þ ln FðxÞ sr1fðyÞ FðxÞ; x < y: ð1:4Þ

Characterization of distributions through conditional expec-tation of record values have been considered among others by Nagraja [5], Franco and Ruiz ([6,7]), Lo´pez-Bla´zquez and Moreno-Rebollo [8], Dembin´ska and Wesolowski [9], Raqab [10], Athar et al. [11], Khan and Alzaid [12] and Wu [13]. In these papers, the authors have assumed the linearity of regression while characterizing the distribution functions.

Khan et al. [14]characterized a family of continuous dis-tributions through difference of two conditional expectations of record values. In this paper, we generalize the result of Khan et al. [14] and have characterized two general forms of distributions through conditional expectation of p th power of difference of functions of two upper record values.

2. Characterization results

Theorem 2.1. Let X be an absolutely continuous random variable with df F(x) and pdf f(x) on the support (a, b). Then, for two values of r and s, 1 6 r< s 6 n,

E½fhðXUðsÞÞ hðXUðrÞÞg p jXUðrÞ¼ x ¼ gr;s;p ¼ 1 ap Cðp þ s rÞ Cðs rÞ ; ð2:1Þ if and only if FðxÞ ¼ 1 eahðxÞ_; _{a >}_0; _ð2:2Þ

where h(x) is a continuous, differentiable, and non-decreasing function of x and p is a positive integer.

Proof. To prove the necessary part, we have for sP r + 1

E½fhðXUðsÞÞ hðXUðrÞÞg p jXUðrÞ¼ x ¼ 1 ðs r 1Þ! Zb x

ðhðyÞ hðxÞÞp½ ln FðyÞ þ ln FðxÞsr1fðyÞ FðxÞdy

¼ 1

ðs r 1Þ! Zb

x

ðhðyÞ hðxÞÞp½ahðyÞ ahðxÞsr1e ahðyÞ eahðxÞah 0 ðyÞdy ¼ a sr ðs r 1Þ! Zb x

ðhðyÞ hðxÞÞpþsrþ1eaðhðyÞhðxÞÞ_h0_ðyÞdy:

Let h(y) h(x) = t, then

E½fhðXUðsÞÞ hðXUðrÞÞgpjXUðrÞ¼ x

¼ a sr ðs r 1Þ! Z 1 0 tpþsr1_eat_dt ¼ a sr ðs r 1Þ! ðp þ s r 1Þ! apþsr ¼ 1 apðp þ s r 1Þðp þ s r 2Þ ðs rÞ: This proves the necessary part.

To prove the sufﬁciency part, suppose

E½fhðXUðsÞÞ hðXUðrÞÞg p jXUðrÞ¼ x ¼ gr;s;p or, 1 ðs r 1Þ! Z b x ðhðyÞ hðxÞÞp_{½ ln FðyÞ þ ln FðxÞ}sr1 fðyÞdy ¼ gr;s;pFðxÞ:

Differentiating both sides w.r.t. x, we get

ph0ðxÞ 1 ðsr1Þ! Rb xðhðyÞ hðxÞÞ p1 ½ ln FðyÞ þ ln FðxÞsr1fðyÞdy _FðxÞfðxÞ 1 ðsr2Þ! Rb xðhðyÞ hðxÞÞ p ½ ln FðyÞ þ ln FðxÞsr2fðyÞdy ¼ fðxÞgr;s;p or, ph0ðxÞ 1 ðsr1Þ! Rb xðhðyÞ hðxÞÞ p1_{½ ln FðyÞ þ ln FðxÞ}s_{r1 fðyÞ} FðxÞdy þfðxÞ FðxÞ 1 ðsr2Þ! Rb xðhðyÞ hðxÞÞ p ½ ln FðyÞ þ ln FðxÞsr2 fðyÞ FðxÞdy ¼_FðxÞfðxÞg_r;s;p or, ph0ðxÞg_r;s;p1þfðxÞ FðxÞgrþ1;s;p¼ fðxÞ FðxÞgr;s;p implying fðxÞ FðxÞ¼ ah 0_ðxÞ as g_r;s;p1¼ 1 ap1 Cðp þ s r 1Þ Cðs rÞ and gr;s;p grþ1;s;p¼ p ap Cðp þ s r 1Þ Cðs rÞ Hence the theorem. h

The case when p = 1,

which is the special case of Theorem 2.1, have been treated in Khan et al.[14].

Corollary 2.1. Let X be an absolutely continuous random variable with df F(x) and pdf f(x) on the support (a, b). Then, for r< s,

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E½fhðXUðsÞÞ hðXUðrÞÞg 2 jXUðrÞ¼ x ¼ 1 a2ðs rÞðs r þ 1Þ ð2:3Þ if and only if FðxÞ ¼ 1 eahðxÞ_; _{a >}_0; _ð2:4Þ

where h(x) is a continuous, differentiable, and non-decreasing function of x.

Proof. The proof is obvious at p = 2 in Theorem 2.1. h

Theorem 2.2. Let X be an absolutely continuous random variable with df F(x) and pdf f(x) on the support (a, b). Then, for two consecutive values of r and s, 1 6 r< s 6 n,

E½fhðXUðsÞÞ hðXUðrÞÞgpjXUðrÞ¼ x ¼ a

Xp j¼0 p j ðhðxÞÞpj b a j ð2:5Þ if and only if FðxÞ ¼ ½ahðxÞ þ bc ; a –0; ð2:6Þ where a_¼X p i¼0 ð1Þiþp p i _c cþ i sr

and h(x) is a continuous and differentiable function of x.

Proof. To prove the necessary part, we have

E½fhðXUðsÞÞ hðXUðrÞÞg p jXUðrÞ¼ x ¼ 1 ðs r 1Þ! Z b x

ðhðyÞ hðxÞÞp½ ln FðyÞ þ ln FðxÞsr1 fðyÞ FðxÞdy ¼ 1 ðs r 1Þ! Z b x ðhðyÞ hðxÞÞp lnFðxÞ FðyÞ sr1 fðyÞ FðxÞdy ¼ 1 ðs r 1Þ! Z b x ðhðyÞ hðxÞÞp ln ahðxÞ þ b ahðyÞ þ b c sr1 ahðyÞ þ b ahðxÞ þ b c ach0ðyÞ ðahðyÞ þ bÞdy: Let ln ahðxÞ þ b ahðyÞ þ b c ¼ t or ahðxÞ þ b ahðyÞ þ b c ¼ et implying hðyÞ hðxÞ ¼ 1 aðahðxÞ þ bÞð1 e t=c_Þ or ðhðyÞ hðxÞÞp¼ð1Þ p ap ðahðxÞ þ bÞ p ð1 et=c_Þp_: Then, we have

E½fhðXUðsÞÞ hðXUðrÞÞgpjXUðrÞ¼ x

¼ ð1Þ p ap_{ðs r 1Þ!}ðahðxÞ þ bÞ pZ 1 0 ð1 et=c_Þp_tsr1_et_dt ¼ ð1Þ p ap_{ðs r 1Þ!}ðahðxÞ þ bÞ pZ 1 0 Xp i¼0 p i ! ð1Þieit=ctsr1etdt ¼ ð1Þ p ap_{ðs r 1Þ!}ðahðxÞ þ bÞ pX p i¼0 p i ! ð1Þi Z 1 0 tsr1_etði cþ1Þ_dt ¼ ð1Þ p ap_{ðs r 1Þ!}ðahðxÞ þ bÞ pX p i¼0 p i ! ð1Þiðs r 1Þ! ðiþc cÞ sr ¼ð1Þ p ap ðahðxÞ þ bÞ pX p i¼0 p i ! ð1Þi c iþ c sr :

This proves the necessary part. To prove sufﬁciency part, let

nr;s;pðxÞ ¼ E½fhðXUðsÞÞ hðXUðrÞÞgpjXUðrÞ¼ x

¼ aX p j¼0 p j ðhðxÞÞpj b a j :

Now proceeding on the lines of Theorem 2.1, we have

fðxÞ FðxÞ¼

n0_r;s;pðxÞ þ ph0ðxÞnr;s;p1ðxÞ

nr;s;pðxÞ nrþ1;s;pðxÞ

: ð2:7Þ

Table 3.1 Examples based on the df F(x) = 1 eah(x)

. Distribution F(x) a h(x) Exponential 1 ehx _h _x 0 < x <1 Weibull 1 ehxp h xp 0 < x <1 Pareto 1 x a h h log x a a < x <1 Lomax 1 1 þ x a p p log½1 þ x a 0 < x <1 Gompertz 1 exp k lðe lx_1Þ h i k l e lx 1 0 < x <1 Beta of I kind 1 (1 x)h h log(1 x) 0 < x < 1

Beta of II kind 1 (1 + x)1 ₁ _{log(1 + x)} 0 < x <1

Extreme value I 1 exp[ex_] ₁ _ex

0 < x <1 Log logistic 1 (1 + hxp

)1 1 log(1 + h xp) 0 < x <1

Burr type IX 1hcfð1þe₂xÞk1gþ 1i1 1 loghcfð1þe₂xÞk1gþ 1i 1 < x < 1

Burr type XII 1 (1 + hxp

)m m log(1 + h xp) 0 < x <1

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Consider n0_r;s;pðxÞ þ ph0ðxÞnr;s;p1ðxÞ ¼ pah0ðxÞ ð1Þp ap ðahðxÞ þ bÞp1X p i¼0 p i ! ð1Þi c iþ c sr þ ph0ðxÞð1Þ p1 ap1 ðahðxÞ þ bÞ p1Xp1 i¼0 p 1 i ! ð1Þi c iþ c sr ¼ ph0_ðxÞð1Þ p ap1 ðahðxÞ þ bÞ p1 X p i¼0 p i ! ð1Þi c iþ c sr X p1 i¼0 p 1 i ! ð1Þi c iþ c sr " # ¼ ph0_ðxÞð1Þ p ap1 ðahðxÞ þ bÞ p1 X p i¼0 p i ! ð1Þi c iþ c sr X p1 i¼0 p i ! ðp iÞ p ð1Þ i c iþ c sr " # ¼ ph0_ðxÞð1Þ p ap1 ðahðxÞ þ bÞ p1 X p i¼0 p i ! ð1Þi c iþ c sr X p1 i¼0 p i ! ð1Þi c iþ c sr " þ1 p Xp1 i¼0 p i ! ð1Þii c iþ c sr# ¼ ph0ðxÞð1Þ p ap1 ðahðxÞ þ bÞ p11 p Xp i¼0 p i ! ð1Þii c iþ c sr

Table 3.2 Examples based on the df F(x) = 1 [ah(x) + b]c_.

Distribution F(x) a b c h(x) Power function apxp ap 1 1 xp 0 < x 6 a Pareto 1 ap xp aq 0 p/q xq, q„ 0 a 6 x<1 Beta of I Kind 1 (1 x)p ₁ ₀ _p/q ₍₁ x)q_{, q} „ 0 0 6 x 6 1 1 1 p x Weibull 1 ehxp 1 0 h/q eqxp 0 6 x <1 h/c 1 1 xp Inverse Weibull ehxp 1 1 1 ehxp 0 6 x <1 Burr type II [1 + ex]k 1 1 1 (1 + ex)k 1 < x < 1

Burr type III [1 + xc]k 1 1 1 (1 + xc)k

0 6 x <1 Burr type IV ½1 þ ðcx xÞ 1=c k 1 1 1 ½1 þ ðcx xÞ 1=c k 0 6 x < c

Burr type V [1 + c etanx]k 1 1 1 [1 + c etan x]k

p/2 6 x 6 p/2

Burr type VI [1 + c eksinh x]k 1 1 1 [1 + c eksinh x]k

1 < x < 1

Burr type VII 2k(1 + tanhx)k

2k 1 1 [1 + tanh x]k

1 < x < 1

Burr type VIII 2

ptan1ex k 2 p k 1 1 (tan1ex)k 1 < x < 1

Burr type IX 1hcfð1þex₂Þk1gþ2i1 c/2 1 c/2 1 (1 + ex₎1

1 < x < 1 Burr type X ð1 ex2 Þk 1 1 1 ð1 ex2 Þk 0 < x <1 Burr type XI x1 2psin 2px k 1 1 1 x1 2psin 2px k 0 6 x 6 1

Burr type XII 1 (1 + hxp₎m _h ₁

m xp 0 6 x <1 Cauchy 1 2þ1ptan1x 1p 12 1 tan 1_x 1 < x < 1

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and nr;s;pðxÞ nrþ1;s;pðxÞ ¼ ð1Þp ap ðahðxÞ þ bÞ pX p i¼0 p i ! ð1Þi c iþ c sr ð1Þ p ap ðahðxÞ þ bÞ pX p i¼0 p i ! ð1Þi c iþ c sr1 ¼ð1Þ p ap ðahðxÞ þ bÞ pX p i¼0 p i ! ð1Þi c iþ c sr 1iþ c c ¼ ð1Þ p cap ðahðxÞ þ bÞ pX p i¼0 p i ! ð1Þi i c iþ c sr :

Therefore in view of(2.7), we get

fðxÞ FðxÞ¼ pach0ðxÞ ðahðxÞ þ bÞ 1 p Pp i¼0 p i ð1Þii c iþc sr Pp i¼0 p i ð1Þi i c iþc sr ¼ ach0ðxÞ ahðxÞ þ b:

Hence the theorem. h

Corollary 2.2. Under the condition stated in Theorem 2.2

E½hðXUðsÞÞjXUðrÞ¼ x ¼ ahðxÞ þ b ð2:8Þ

where a¼ c 1þ c sr and b¼ b a ð1 a_Þ if and only if FðxÞ ¼ ½ahðxÞ þ bc; a –0:

Proof. (2.8) can be proved in view of Theorem 2.2 at p= 1. h

3. Examples

SeeTables 3.1 and 3.2.

Acknowledgement

The authors acknowledge with thanks to both the referees and the Editor for their fruitful suggestions and comments

which led the overall improvement in the manuscript. Authors are also thankful to Professor A.H. Khan, Aligarh Muslim University, Aligarh, who helped in preparation of this manuscript.

References

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[2]B.C. Arnold, N. Balakrishnan, H.N. Nagraja, Records, John Wiley, New York, 1998.

[3]M. Ahsanullah, Record Values – Theory and Applications, University Press of America, Lanham, Maryland, 2004. [4]M. Ahsanullah, Record Statistics, Nova Science Publishers,

New York, 1995.

[5]H.N. Nagaraja, Some characterizations of continuous distributions based on regressions of adjacent order statistics and record values, Sankhya Ser. A 50 (1988) 70–73. [6]M. Franco, J.M. Ruiz, On characterization of continuous distributions by conditional expectation of record values, Sankhya Ser. A 58 (1996) 135–141.

[7]M. Franco, J.M. Ruiz, On characterization of distributions by expected values of order statistics and record values with gap, Metrika 45 (1997) 107–119.

[8]F. Lo´pez-Bla´zquez, J.L. Moreno-Rebollo, A characterization of distributions based on linear regression of order statistics and record values, Sankhya Ser. A 59 (1997) 311–323.

[9]A. Dembin´ska, J. Wesolowski, Linearity of regression for non-adjacent record values, J. Statist. Plann. Inference 90 (2000) 195– 205.

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[11]H. Athar, M. Yaqub, H.M. Islam, On characterization of distributions through linear regression of record values and order statistics, Aligarh J. Statist. 23 (2003) 97–105.

[12]A.H. Khan, A.A. Alzaid, Characterization of distributions through linear regression of non-adjacent generalized order statistics, J. Appl. Statist. Sci. 13 (2004) 123–136.

[13]J.W. Wu, On characterizing distributions by conditional expectation of functions of non-adjacent record values, J. Appl. Statist. Sci. 13 (2004) 137–145.

[14]A.H. Khan, M. Faizan, Z. Haque, Characterization of continuous distributions through record statistics, Commun. Korean Math. Soc. 25 (3) (2010) 485–489.