On characterization of continuous distributions via conditional expectation of non-adjacent order statistics

Selçuk J. Appl. Math. Selçuk Journal of Vol. 14. No. 1. pp. 11-20, 2013 Applied Mathematics

On Characterization of Continuous Distributions via Conditional Ex-pectation of Non-adjacent Order Statistics

A. H. Khan, M. Faizan1_{, Ziaul Haque}

1_{Department of Statistics and Operations Research, Aligarh Muslim University,}

Aligarh-202 002. India.

e-mail:m dfaizan02@ redi¤m ail.com

Received Date: February 21, 2011 Accepted Date: December 06, 2012

Abstract. In this paper, families of continuous probability distributions have been characterized through conditional expectations, conditioned on a non-adjacent order statistic. Further, some of its important deductions are also discussed.

Key words: Characterization; continuous distributions; conditional expecta-tion; order statistics.

AMS Classi…cation: 62G30, 62E10. 1. Introduction

Let X1; X2; : : : ; Xnbe a random sample of size n from a continuous population having the probability density function (pdf ) f (x) and the distribution function (df ) F (x) and let X1:n < X2:n < : : : < Xn:n be the corresponding order statistics. Then the conditional pdf of Xs: n given Xr: n= x, 1 r < s n, is (David and Nagaraja, 2003)

(1.1) fsjr(yjx) = (n r)! (s r 1)!(n s)! [F (y) F (x)]s r 1[1 F (y)]n s [1 F (x)]n r f (y); x < y and the conditional pdf of Xr: n given Xs : n = y, 1 r < s n is (David and Nagaraja, 2003) (1.2) frjs(xjy) = (s 1)! (r 1)!(s r 1)! [F (x)]r 1[F (y) F (x)]s r 1 [F (y)]s 1 f (x) ; x < y 1_{Corresponding author}

Conditional expectations of order statistics are extensively used in characterizing the continuous probability distributions. Various approaches are available in the literature. For a detailed survey one may refer to Ferguson (1967), Khan and Abu-Salih (1989), Khan and Abouammoh (2000), Khan et al. (2009) amongst others.

In this paper, we have characterized general forms of distributions by considering conditional expectation of functions of order statistics when the conditioning is on any order statistic, not necessarily the adjacent one.

2. Characterization Theorems

Theorem 2.1. Let X be an absolutely continuous random variable with the df F (x) and the pdf f (x) on the support( ; ), where and may be …nite or in…nite. Then for 1 m < r < s n,

(2.1) E[h (Xs : n)jXm : n = x] = asj rE[h (Xr : n)jXm : n= x] + bsj r if and only if

(2.2) F (x) = 1 [a h(x) + b]c_{; x 2 ( ; )}

where a_{sj r} =Qs 1_{j=r c(n j)+1}c (n j) , b_{sj r} = _ab(1 a_{sj r}) and h(x) is a monotonic and di¤erentiable function such that F (x) is a df for …xed a; b and c.

Proof. In view of Khan and Abouammoh (2000), we have for the df F (x) = 1 [a h(x) + b]c E[h (Xs : n)jXm : n= x] = asj mh(x) + bsj m (2.3) = a_{sj m} h(x) +b a b a where a_sjm=Qs 1_{j=m c(n j)+1}c (n j) = r_Y1 j=m c (n j) c(n j) + 1 s 1_Y j=r c (n j) c(n j) + 1 = arjmasjr and b_{sj m}= b a(1 asj m) Therefore, E[h (Xs : n)jXm : n= x] = asjr arjm h(x) + b a b a + b aasjr b a = a_sjr[a_rjmh(x) + b_rjm] + b_sjr = a_{sj r}E[h (Xr : n)jXm : n= x] + bsj r

For the su¢ ciency part, we have (n m)!

(s m 1)!(n s)! Z

x

h(y)[F (y) F (x)]s m 1[1 F (y)]n sf (y) dy

= a_{sj r} (n m)! (r m 1)!(n r)! Z x h(y)[F (y) F (x)]r m 1 (2.4) [1 F (y)]n rf (y) dy + b_{sj r} [1 F (x)]n m Di¤erentiating (r m) times both the sides of (2.4) w.r.t. x, we get

(n r)! (s r 1)!(n s)!

Z x

h(y)[F (y) F (x)]s r 1 [1 F (y)]n sf (y) dy

= a_{sj r}h(x) [1 F (x)]n r + b_{sj r} [1 F (x)]n r or, (n r)! (s r 1)!(n s)! [1 F (x)]n r Z x h(y)[F (y) F (x)]s r 1 (2.5) [1 F (y)]n sf (y) dy = a_{sj r}h(x) + b_{sj r} = gsj r(x) where, E[h (Xs : n)jXr : n= x] = gsj r(x) Using the result Khan et al. (2006, 2007)

E[h (Xs : n)jXr : n= x] = gsj r(x) we have, (2.6) F (x) = e RxA(t) dt where, A(t) = g 0 sjr(t) (n r) [gsjr(t) gsjr+1(t)] = a c h0(t) [a h(t) + b] Thus, F (x) = e Rx a c h0 (t) [a h(t)+b]dt and F (x) = 1 [a h(x) + b]c and hence the Theorem.

Table 2.1. Examples based on the df F (x) = 1 [ah(x) + b]c

2:jpg

Abouam-moh (2000).

Further when a = a_{c; b = 1; c ! 1} then for the df F (x) = 1 e a h(x)_{; a 6= 0}

a_{sj r}= 1 and b_{sj r}= 1 a

Ps 1 j=r (n j)1 , as obtained by Khan et al. (2009).

Theorem 2.2. Under the conditions as given in the Theorem 2.1, for 1 m < r < s n,

(2.7) E[h (Xs : n) h (Xr : n)jXm : n= x] = E[h (Xs : n) h (Xr : n)] if and only if

(2.8) F (x) = 1 e a h(x)_{; a 6= 0}

Proof. It can be seen (Khan et al., 1983) that for 1 r < s n, (2.9) E[h(Xs : n) h(Xs 1 : n)] = n s 1 Z h0(x) [F (x)]s 1[1 F (x)]n s+1dx Therefore, for F (x) = e a h(x)_{, R.H.S. of (2.7) reduces to}

E[h(Xs : n) h(Xr : n)] = s X j=r+1 E[h(Xj : n) h(Xj 1 : n)] =1 a s 1 X j=r 1 (n j) and the L.H.S. is E[h (Xs : n) h (Xr : n)jXm : n= x] = (asj m arj m) h(x) + (bsj m brj m) =1 a s 1 X j=m 1 (n j) 1 a r 1 X j=m 1 (n j) =1 a s 1 X j=r 1 (n j) Hence the necessary part is proved.

For the su¢ ciency part, let E[h (Xs : n) h (Xr : n)] = c , independent of x, therefore, we have

(n m)! (s m 1)!(n s)!

Z x

(n m)! (r m 1)!(n r)!

Z x

h(y)[F (y) F (x)]r m 1[1 F (y)]n rf (y) dy

(2.10) = c [1 F (x)]n m

Di¤erentiating (r m) times both the sides of (2.10) w.r.t. x, we get (n r)! (s r 1)!(n s)! [1 F (x)]n r Z x h(y)[F (y) F (x)]s r 1 [1 F (y)]n sf (y) dy = h(x) + c or, gsj r(x) = h(x) + c Using the result Khan et al. (2006, 2007)

E[h (Xs : n)jXr : n= x] = gsj r(x) we get, (2.11) F (x) = e RxA(t) dt where, A(t) = g 0 sjr(t) (n r) [gsjr(t) gsjr+1(t)] = a h0(t) Thus, F (x) = e Rx a h0(t) dt and F (x) = 1 e a h(x)_{; a 6= 0} and hence the Theorem.

Remark 2.2. Examples based on the df F (x) = 1 e a h(x)_{; a 6= 0 may be} obtained from Khan et al. (2009).

Theorem 2.3. Under the conditions as given in the Theorem 2.1, for 1 m < r < s n,

(2.12) E[h (Xm : n)jXs : n = y] = amj rE[h (Xr : n)jXs : n = y] + bmj r if and only if

where

a_{mj r}=Qr_j=m1 _{1+c j}c j and b_{mj r} = b

a(1 amj r). Proof. Proceeding as in the Theorem 2.1, we get

E[h (Xm : n)jXs : n = y] = amj rE[h (Xr : n)jXs : n = y] + bmj r For the su¢ ciency part, we have

(s 1)! (s m 1)!(m 1)! Z y h(x)[F (y) F (x)]s m 1[F (x)]m 1f (x) dx = a_{mj r} (s 1)! (s r 1)!(r 1)! Z y h(x)[F (y) F (x)]s r 1[F (x)]r 1f (x) dx (2.14) +b_{mj r} [F (y)]s 1

Di¤erentiating (s r) times both the sides of (2.14) w.r.t. y, we get (r 1)!

(r m 1)!(m 1)! Z y

h(x)[F (y) F (x)]r m 1 [F (x)]m 1f (x) dx = a_{mj r}h(y) [F (y)]r 1 + b_{mj r} [F (y)]r 1

= gmj r(y) where,

E[h (Xm : n)jXr : n= y] = gmj r(y)

Using Corollary 2.2 of Khan et al. (2006), we note that E[h (Xm : n)jXr : n= y] = gmj r(y) implies (2.15) F (y) = e R y A(t) dt where A(t) = g 0 mjr(t) (r 1) [gmjr 1(t) gmjr(t)] = a c h0(t) [a h(t) + b] Thus, F (y) = e R y a c h0 (t) [a h(t)+b]dt and F (x) = [a h(x) + b]c_{; x 2 ( ; )}

and hence the Theorem.

Table 2.2. Examples based on the df F (x) = [ah(x) + b]c

Remark 2.3. At s = r, we get the result as obtained by Khan and Abouammoh (2000).

Theorem 2.4. Under the conditions as given in the Theorem 2.1, for 1 m < r < s n, (2.16) E[h (Xr : n) h (Xm : n)jXs : n= y] = 1 a r 1 X j=m 1 j if and only if (2.17) F (x) = e a h(x)_{; a 6= 0}

Proof. It can be seen that (Khan and Abouammoh, 2000) for 1 < r < s n E[h (Xr : n) h (Xr 1 : n)jXs : n = y] = s 1 r 1 1 [F (y)]s 1 Z y h0(x) [F (x)]r 1[F (y) F (x)]s rdx E[h (Xr : n) h (Xm : n)jXs : n = y] = r_Xm 1 i=0 E[h (Xr i : n) h (Xr i 1 : n)jXs : n= y] Therefore; = r_Xm 1 i=0 s 1 r i 1 1 [F (y)]s 1 Z y h0(x) [F (x)]r i 1[F (y) F (x)]s r+idx

= 1 a r X j=m+1 s 1 j 1 1 [F (y)]s 1 Z y [F (x)]j 2[F (y) F (x)]s jf (x) dx; as f (x) = a h0_{(x) F (x)} = 1 a r 1 X j=m 1 j

Now to prove that (2.16) implies (2.17), we have for c0=_a1 Pr_j=m1 1_j.

(s 1)! (s r 1)!(r 1)! Z y h(x)[F (y) F (x)]s r 1[F (x)]r 1f (x) dx (s 1)! (s m 1)!(m 1)! Z y h(x)[F (y) F (x)]s m 1[F (x)]m 1f (x) dx (2.18) = c0[F (y)]s 1

Di¤erentiate (s r) times both the sides of (2.18) w.r.t. y, to get (r 1)!

(r m 1)!(r 1)! [F (y)]r 1 Z y

h(x)[F (y) F (x)]r m 1

(2.19) [F (x)]m 1f (x) dx = h(y) + c0 = gmj r(y) Using the result (Khan et al., 2006), we get

F (x) = e a h(x)_{; a 6= 0} and hence the Theorem.

Remark 2.4. At s = r, we get E[h (Xm : n)jXs : n = y] = h(y) + 1 a s 1 X j=m 1 j as obtained by Khan and Abouammoh (2000).

Remark 2.5. Examples based on the df F (x) = e a h(x)_{; a 6= 0 may be} obtained from Khan and Abu-Salih (1989).

Conclusion. Conditional expectation of order statistics conditioned on non-adjacent order statistics are used to characterize the families of distributions F (x) = 1 [a h(x) + b]c and F (x) = [a h(x) + b]c where h(x) is a monotonic and di¤erentiable function such that F (x) is the distribution function for …xed a; b and c. Also conditional expectation of the di¤erence of two order statistics

conditioned on non-adjacent order statistic are considered to characterize the df F (x) = 1 e a h(x)and F (x) = e a h(x). The speci…c distributions for which the results are true are given by proper choice of a; b, c and h(x).

3. Acknowledgments

The authors are thankful to the referees for their valuable comments that im-proved the original version of the manuscript.

References

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