- Email: [email protected]
Characterization of distributions by conditional expectation of record values
Journal of the Egyptian Mathematical Society (2015) xxx, xxx–xxx
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
ORIGINAL ARTICLE
Characterization of distributions by conditional expectation of record values A.H. Khan, Ziaul Haque *, Mohd. Faizan Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh 202002, India Received 1 February 2014; revised 24 July 2014; accepted 3 August 2014
KEYWORDS Characterization; Continuous distributions; Conditional expectation; Record values
Abstract A family of continuous probability distributions has been characterized by two conditional expectations of record statistics conditioned on a non-adjacent record value. Besides various deductions, this work extends the result of Lee [8] in which Pareto distribution has been characterized. 2010 MATHEMATICS SUBJECT CLASSIFICATION:
62E10; 62G30; 60E05
ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction Characterizations of distributions through conditional expectations of record values have been considered among others by Nagaraja [1], Franco and Ruiz [2], Wu and Lee [3], Raqab [4], Athar et al. [5], Gupta and Ahsanullah [6]. Let X1, X2, . . . be a sequence of independent, identically distributed continuous random variables with the distribution function (df) F(x) and the probability density function (pdf) f(x). Let Xu(s) be the s-th upper record value, then the conditional pdf of Xu(s) given Xu(r) = x, 1 6 r < s is Ahsanullah [7]
fðXuðsÞ jXuðrÞ ¼ xÞ ¼
sr1 fðyÞ 1 ln FðyÞ þ ln FðxÞ ; C ðs rÞ FðxÞ
x < y;
ð1:1Þ
where FðxÞ ¼ 1 FðxÞ. Lee [8] has characterized Pareto distribution by conditional expectation of two records Xu(s) and Xu(r) conditioned on Xu(m) for all s > r P m, where s = r + 1, r + 2 and r + 3. In this paper we have characterized a general class of distributions FðxÞ ¼ ½ahðxÞ þ bc by the conditional expectation of Xu(s) and Xu(r) conditioned on Xu(m) for all s > r P m, thus extending the results of Lee [8]. 2. Characterization results
* Corresponding author. E-mail address: [email protected] (Z. Haque). Peer review under responsibility of Egyptian Mathematical Society.
Production and hosting by Elsevier
Theorem 2.1. Let X be an absolutely continuous random variable with the df F(x) and the pdf f(x) on the support (a, b), where a and b may be finite or infinite. Then for m6r
http://dx.doi.org/10.1016/j.joems.2014.08.005 1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. Please cite this article in press as: A.H. Khan et al., Characterization of distributions by conditional expectation of record values, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.005
2
A.H. Khan et al.
E h XuðsÞ jXuðmÞ ¼ x ¼ a E h XuðrÞ jXuðmÞ ¼ x þ b
ð2:1Þ
Similarly, differentiating (r m 1) times both the sides w.r.t. x, we get
ð2:2Þ
1 Cðs rÞ
if and only if FðxÞ ¼ ½ahðxÞ þ bc ; sr c where a ¼ cþ1 and b ¼ ba ð1 a Þ.
where, sm c a1 ¼ cþ1
b
x
sr1 fðyÞ hðyÞ ln FðyÞ þ ln FðxÞ dy FðxÞ
¼ a hðxÞ þ b ¼ gsjr ðxÞ: Using the result (Khan et al. [9]),
Proof. In view of the Athar et al. [5], we have E h XuðsÞ jXuðmÞ ¼ x ¼ a1 hðxÞ þ b1 ;
Z
ð2:3Þ
E h XuðsÞ jXuðrÞ ¼ x ¼ gsjr ðxÞ we get,
b b1 ¼ 1 a1 a
and
and E h XuðrÞ jXuðmÞ ¼ x ¼ a2 hðxÞ þ b2
FðxÞ ¼ e
b2 ¼
and
a
AðtÞdt
where ð2:4Þ AðtÞ ¼
where rm c a2 ¼ cþ1
Rx
b 1 a2 : a
g0sjr ðtÞ gsjr ðtÞ gsjrþ1 ðtÞ
¼
ach0 ðtÞ and lim x!b ½ahðtÞ þ b
Z
x
AðtÞdt ¼ 1: a
Thus,
Using (2.3) and (2.4), it is easy to establish (2.1).
FðxÞ ¼ ½ahðxÞ þ bc
For sufficiency part, we have 1 Cðs mÞ
Z
b
and hence the theorem.
sm1 hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy
x
1 ¼a C ðr m Þ
Z
b
rm1 hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy
x
þ b FðxÞ
ð2:5Þ
Differentiate both the sides of (2.5) w.r.t. x, to get Z
sm2 fðxÞ hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy FðxÞ x Z rm2 ðr m 1Þ b ¼ a hðyÞ ln FðyÞ þ ln FðxÞ Cðr mÞ x fðxÞ fðyÞdy b fðxÞ: FðxÞ R vðxÞ after noting that if B ¼ uðxÞ fðx; yÞdy then
ðs m 1Þ Cðs mÞ
Z
vðxÞ uðxÞ
@fðx; yÞ dy: @x
Therefore,
¼ a
Z
b
sm2 hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy
x
1 Cðr m 1Þ
þ b FðxÞ:
Remark 2.1. At r = m, h(x) = x, we get the result as obtained by Franco and Ruiz [2,10], Athar et al. [5], Ahsanullah and Wesolowski [11], Dembin´ska and Wesolowski [12], Khan and Alzaid [13]. Remark 2.2. Lee [8] has obtained characterization result for Pareto distribution
b
@B @v @u ¼ fðx; vÞ fðx; uÞ þ @x @x @x
1 Cðs m 1Þ
h
Z x
b
rm2 hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy
FðxÞ ¼ xh ;
x > 1;
h > 0;
h – 1;
which can be obtained by putting a = 1, b = 0, c = h, h(x) = x at s = r + 1, r + 2 and r + 3 in the Theorem 2.1. Remark 2.3. At a ¼ ac ; b ¼ 1; c ! 1 a ¼ 1;
b ¼
ðs rÞ ; a
FðxÞ ¼ eahðxÞ ; a > 0, reduces to the result as obtained by Khan et al. [14]. 3. Examples based on the distribution function F(x) = 1 [a h(x) + b]c Proper choice of a, b and h(x) characterize the distributions as given below:
Please cite this article in press as: A.H. Khan et al., Characterization of distributions by conditional expectation of record values, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.005
Characterization of distributions of Record Values
Distribution Power function Pareto Beta of the first kind Weibull Inverse Weibull Burr type II Burr type III Burr type IV Burr type V Burr type VI Burr type VII Burr type VIII Burr type X Burr type XI Burr type XII Cauchy
F(x)
3
a
p p
a x 0
1 ex
k
0 < x1 < 1 k x 2p sin2px 06x61 1 (1 + hxp)m 06x<1 1 1 1 2 þ p tan x 1 < x < 1
b
c
h(x)
a
1
1
xp
a1 aq 1 1 1
0 0 0 1 0
p p/q p/q p h/q
x, p „ 1 xq, q > 0,p „ q (1 x)q, q > 0 x p eqx ; q > 0
1
1
1
ehx
1
1
1
(1 + ex)k
1
1
1
(1 + xc)k
1
1
1
1
1
1
[1 + c etanx]k
1
1
1
[1 + c eksinhx]k
2k
1
1
[1 + tanhx]k
k p2
1
1
1
1
1
1
1
1
h
1
m
xp, m „ 1
p1
1 2
1
tan1x
p
Acknowledgements Authors are thankful to anonymous Reviewer and Editor for their comments and suggestions, which resulted in an improvement in the presentation of this manuscript.
[7]
References
[9]
[1] H.N. Nagaraja, On a characterization based on record values, Aust. J. Statist. 19 (1977) 70–73. [2] M. Franco, J.M. Ruiz, On characterizations of distributions by expected values of order statistics and record values with gap, Metrika 45 (1997) 107–119. [3] J.W. Wu, W.C. Lee, On characterizations of generalized extreme values, power function, generalized Pareto and classical Pareto distributions by conditional expectation of record values, Statist. Papers 42 (2001) 225–242. [4] M.Z. Raqab, Characterizations of distributions based on the conditional expectation of record values, Statist. Dec. 20 (2002) 309–319. [5] H. Athar, M. Yaqub, H.M. Islam, On characterizations of distributions through linear regression of record values and order statistics, Aligarh J. Statist. 23 (2003) 97–105. [6] R.C. Gupta, M. Ahsanullah, Some characterization results based on the conditional expectation of a function of
[8]
[10]
[11] [12]
[13]
[14]
p
h
1þ
cx1=c ik x
tan1 e x
k
2 k 1 ex k 1 x 2p sin2px
non-adjacent order statistic (record value), Ann. Inst. Statist. Math. 56 (2004) 721–732. M. Ahsanullah, Record Statistics, Nova Science Publishers, New York, 1995. M.Y. Lee, Characterizations of the Pareto distribution by conditional expectations of record values, Comm. Korean Math. Soc. 18 (1) (2003) 127–131. A.H. Khan, R.U. Khan, M. Yaqub, Characterization of continuous distributions through conditional expectation of functions of generalized order statistics, J. Appl. Probab. Statist. 1 (2006) 115–131. M. Franco, J.M. Ruiz, On characterization of continuous distributions with adjacent order statistics, Statistics 26 (1995) 375–385. M. Ahsanullah, J. Wesolowski, Linearity of best predictors for non-adjacent record values, Sankhy a, Ser. B 60 (1998) 221–227. A. Dembinska, J. Wesolowski, Linearity of regression for nonadjacent record values, J. Statist. Plann. Inference 90 (2000) 195–205. 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. A.H. Khan, M. Faizan, Z. Haque, Characterization of continuous distributions through record statistics, Comm. Korean Math. Soc. 25 (3) (2010) 485–489.
Please cite this article in press as: A.H. Khan et al., Characterization of distributions by conditional expectation of record values, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.005
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
ORIGINAL ARTICLE
Characterization of distributions by conditional expectation of record values A.H. Khan, Ziaul Haque *, Mohd. Faizan Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh 202002, India Received 1 February 2014; revised 24 July 2014; accepted 3 August 2014
KEYWORDS Characterization; Continuous distributions; Conditional expectation; Record values
Abstract A family of continuous probability distributions has been characterized by two conditional expectations of record statistics conditioned on a non-adjacent record value. Besides various deductions, this work extends the result of Lee [8] in which Pareto distribution has been characterized. 2010 MATHEMATICS SUBJECT CLASSIFICATION:
62E10; 62G30; 60E05
ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction Characterizations of distributions through conditional expectations of record values have been considered among others by Nagaraja [1], Franco and Ruiz [2], Wu and Lee [3], Raqab [4], Athar et al. [5], Gupta and Ahsanullah [6]. Let X1, X2, . . . be a sequence of independent, identically distributed continuous random variables with the distribution function (df) F(x) and the probability density function (pdf) f(x). Let Xu(s) be the s-th upper record value, then the conditional pdf of Xu(s) given Xu(r) = x, 1 6 r < s is Ahsanullah [7]
fðXuðsÞ jXuðrÞ ¼ xÞ ¼
sr1 fðyÞ 1 ln FðyÞ þ ln FðxÞ ; C ðs rÞ FðxÞ
x < y;
ð1:1Þ
where FðxÞ ¼ 1 FðxÞ. Lee [8] has characterized Pareto distribution by conditional expectation of two records Xu(s) and Xu(r) conditioned on Xu(m) for all s > r P m, where s = r + 1, r + 2 and r + 3. In this paper we have characterized a general class of distributions FðxÞ ¼ ½ahðxÞ þ bc by the conditional expectation of Xu(s) and Xu(r) conditioned on Xu(m) for all s > r P m, thus extending the results of Lee [8]. 2. Characterization results
* Corresponding author. E-mail address: [email protected] (Z. Haque). Peer review under responsibility of Egyptian Mathematical Society.
Production and hosting by Elsevier
Theorem 2.1. Let X be an absolutely continuous random variable with the df F(x) and the pdf f(x) on the support (a, b), where a and b may be finite or infinite. Then for m6r
http://dx.doi.org/10.1016/j.joems.2014.08.005 1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. Please cite this article in press as: A.H. Khan et al., Characterization of distributions by conditional expectation of record values, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.005
2
A.H. Khan et al.
E h XuðsÞ jXuðmÞ ¼ x ¼ a E h XuðrÞ jXuðmÞ ¼ x þ b
ð2:1Þ
Similarly, differentiating (r m 1) times both the sides w.r.t. x, we get
ð2:2Þ
1 Cðs rÞ
if and only if FðxÞ ¼ ½ahðxÞ þ bc ; sr c where a ¼ cþ1 and b ¼ ba ð1 a Þ.
where, sm c a1 ¼ cþ1
b
x
sr1 fðyÞ hðyÞ ln FðyÞ þ ln FðxÞ dy FðxÞ
¼ a hðxÞ þ b ¼ gsjr ðxÞ: Using the result (Khan et al. [9]),
Proof. In view of the Athar et al. [5], we have E h XuðsÞ jXuðmÞ ¼ x ¼ a1 hðxÞ þ b1 ;
Z
ð2:3Þ
E h XuðsÞ jXuðrÞ ¼ x ¼ gsjr ðxÞ we get,
b b1 ¼ 1 a1 a
and
and E h XuðrÞ jXuðmÞ ¼ x ¼ a2 hðxÞ þ b2
FðxÞ ¼ e
b2 ¼
and
a
AðtÞdt
where ð2:4Þ AðtÞ ¼
where rm c a2 ¼ cþ1
Rx
b 1 a2 : a
g0sjr ðtÞ gsjr ðtÞ gsjrþ1 ðtÞ
¼
ach0 ðtÞ and lim x!b ½ahðtÞ þ b
Z
x
AðtÞdt ¼ 1: a
Thus,
Using (2.3) and (2.4), it is easy to establish (2.1).
FðxÞ ¼ ½ahðxÞ þ bc
For sufficiency part, we have 1 Cðs mÞ
Z
b
and hence the theorem.
sm1 hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy
x
1 ¼a C ðr m Þ
Z
b
rm1 hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy
x
þ b FðxÞ
ð2:5Þ
Differentiate both the sides of (2.5) w.r.t. x, to get Z
sm2 fðxÞ hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy FðxÞ x Z rm2 ðr m 1Þ b ¼ a hðyÞ ln FðyÞ þ ln FðxÞ Cðr mÞ x fðxÞ fðyÞdy b fðxÞ: FðxÞ R vðxÞ after noting that if B ¼ uðxÞ fðx; yÞdy then
ðs m 1Þ Cðs mÞ
Z
vðxÞ uðxÞ
@fðx; yÞ dy: @x
Therefore,
¼ a
Z
b
sm2 hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy
x
1 Cðr m 1Þ
þ b FðxÞ:
Remark 2.1. At r = m, h(x) = x, we get the result as obtained by Franco and Ruiz [2,10], Athar et al. [5], Ahsanullah and Wesolowski [11], Dembin´ska and Wesolowski [12], Khan and Alzaid [13]. Remark 2.2. Lee [8] has obtained characterization result for Pareto distribution
b
@B @v @u ¼ fðx; vÞ fðx; uÞ þ @x @x @x
1 Cðs m 1Þ
h
Z x
b
rm2 hðyÞ ln FðyÞ þ ln FðxÞ fðyÞdy
FðxÞ ¼ xh ;
x > 1;
h > 0;
h – 1;
which can be obtained by putting a = 1, b = 0, c = h, h(x) = x at s = r + 1, r + 2 and r + 3 in the Theorem 2.1. Remark 2.3. At a ¼ ac ; b ¼ 1; c ! 1 a ¼ 1;
b ¼
ðs rÞ ; a
FðxÞ ¼ eahðxÞ ; a > 0, reduces to the result as obtained by Khan et al. [14]. 3. Examples based on the distribution function F(x) = 1 [a h(x) + b]c Proper choice of a, b and h(x) characterize the distributions as given below:
Please cite this article in press as: A.H. Khan et al., Characterization of distributions by conditional expectation of record values, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.005
Characterization of distributions of Record Values
Distribution Power function Pareto Beta of the first kind Weibull Inverse Weibull Burr type II Burr type III Burr type IV Burr type V Burr type VI Burr type VII Burr type VIII Burr type X Burr type XI Burr type XII Cauchy
F(x)
3
a
p p
a x 0
1 ex
k
0 < x1 < 1 k x 2p sin2px 06x61 1 (1 + hxp)m 06x<1 1 1 1 2 þ p tan x 1 < x < 1
b
c
h(x)
a
1
1
xp
a1 aq 1 1 1
0 0 0 1 0
p p/q p/q p h/q
x, p „ 1 xq, q > 0,p „ q (1 x)q, q > 0 x p eqx ; q > 0
1
1
1
ehx
1
1
1
(1 + ex)k
1
1
1
(1 + xc)k
1
1
1
1
1
1
[1 + c etanx]k
1
1
1
[1 + c eksinhx]k
2k
1
1
[1 + tanhx]k
k p2
1
1
1
1
1
1
1
1
h
1
m
xp, m „ 1
p1
1 2
1
tan1x
p
Acknowledgements Authors are thankful to anonymous Reviewer and Editor for their comments and suggestions, which resulted in an improvement in the presentation of this manuscript.
[7]
References
[9]
[1] H.N. Nagaraja, On a characterization based on record values, Aust. J. Statist. 19 (1977) 70–73. [2] M. Franco, J.M. Ruiz, On characterizations of distributions by expected values of order statistics and record values with gap, Metrika 45 (1997) 107–119. [3] J.W. Wu, W.C. Lee, On characterizations of generalized extreme values, power function, generalized Pareto and classical Pareto distributions by conditional expectation of record values, Statist. Papers 42 (2001) 225–242. [4] M.Z. Raqab, Characterizations of distributions based on the conditional expectation of record values, Statist. Dec. 20 (2002) 309–319. [5] H. Athar, M. Yaqub, H.M. Islam, On characterizations of distributions through linear regression of record values and order statistics, Aligarh J. Statist. 23 (2003) 97–105. [6] R.C. Gupta, M. Ahsanullah, Some characterization results based on the conditional expectation of a function of
[8]
[10]
[11] [12]
[13]
[14]
p
h
1þ
cx1=c ik x
tan1 e x
k
2 k 1 ex k 1 x 2p sin2px
non-adjacent order statistic (record value), Ann. Inst. Statist. Math. 56 (2004) 721–732. M. Ahsanullah, Record Statistics, Nova Science Publishers, New York, 1995. M.Y. Lee, Characterizations of the Pareto distribution by conditional expectations of record values, Comm. Korean Math. Soc. 18 (1) (2003) 127–131. A.H. Khan, R.U. Khan, M. Yaqub, Characterization of continuous distributions through conditional expectation of functions of generalized order statistics, J. Appl. Probab. Statist. 1 (2006) 115–131. M. Franco, J.M. Ruiz, On characterization of continuous distributions with adjacent order statistics, Statistics 26 (1995) 375–385. M. Ahsanullah, J. Wesolowski, Linearity of best predictors for non-adjacent record values, Sankhy a, Ser. B 60 (1998) 221–227. A. Dembinska, J. Wesolowski, Linearity of regression for nonadjacent record values, J. Statist. Plann. Inference 90 (2000) 195–205. 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. A.H. Khan, M. Faizan, Z. Haque, Characterization of continuous distributions through record statistics, Comm. Korean Math. Soc. 25 (3) (2010) 485–489.
Please cite this article in press as: A.H. Khan et al., Characterization of distributions by conditional expectation of record values, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.005