On order statistics from nonidentical discrete random variables

Bahadır Yüzbaşı; Yunus Bulut; Mehmet Güngör

doi:10.1515/phys-2016-0022

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On order statistics from nonidentical discrete random variables

Bahadır Yüzbaşı , Yunus Bulut and Mehmet Güngör

Published/Copyright: June 24, 2016

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From the journal Open Physics Volume 14 Issue 1

Abstract

In this study, pf and df of single order statistic of nonidentical discrete random variables are obtained. These functions are also expressed in integral form. Finally, pf and df of extreme of order statistics of random variables for the nonidentical discrete case are given.

Keywords: Order statistics; discrete random variable; probability function; distribution function; permanent

PACS: 05.10.-a; 02.50.-r

1 Introduction

Some recurrence relations and identities on the distribution function (df) and probability density function (pdf) of order statistics, from independent and identically distributed (iid) random variables, are provided in the literature by many authors including Arnold et al. [1], David [2], Balasubramanian and Beg [3], and Reiss [4]. Furthermore, Arnoldet al. [1], Gan and Bain [5], David [2], and Khatri [6] established the df and probability function (pf) of order statistics of iid discrete random variables. Corley [7] defined a multivariate generalization of order statistics of continuous multivariate random variables. Goldie and Maller [8] obtained identities for the densities of order statistics of random variables by using the different operators for the iid case. The relation between the probabilities of independent, but not necessarily identically distributed (innid) and the probabilities of iid random vectors is expressed by Guilbaud [9].

Cao and West [10] established the recurrence relations for the df's of order statistics of random variables in case innid. Vaughan and Venables [11], Balakrishnan [12], and Bapat and Beg [13] denoted the pdf and df of order statistics of random variables with permanents in case innid. Childs and Balakrishnan [14] expressed the pdf of X_r:n+1 by adding an another random variable to X₁, X₂,..., X_n using multinomial arguments. Also, Balasubramanianet al. [15] obtained expressions related to pdf and df of order statistics from innid by different methods. Beg [16] established identities and recurrence relations for the moments of order statistics via permanents when the random variables are innid. Crameret al...[17] obtained the identities for the pdf and the df for using both Ryser's method and permanents. Balasubramanianet al..[18] derived the distributions of the single order statistic by distribution functions of the extreme order statistics of subsets of innid random variables X₁, X₂,..., X_n. Furthermore, Balasubramanianet al. [19] generalized the results in [18].

Balakrishnan [20] obtained the relations for order statistics of random variables in continuous and discrete case.

Nagaraja [21, 22] expressed the results about order statistics of random variables for discrete case.

If a₁, a₂,... are defined as column vectors, then the matrix obtained by taking j₁, j₂,... copies of a₁, a₂,..., respectively, can be defined as

[a1j1 a2j2...],

and perA is called the permanent of A which is a square matrix. The definition of perA is similar to determinants, but where every term in the expansion is positive.

Let X₁, X₂,..., X_n be innid discrete random variables and X_1:n ≤ X_2:n ≤ ... ≤ X_n:n be order statistics of the nX_k's. Let F_k and f_k be df and pf of X_k, respectively.

Some familiar applications of order statistics are given as follows:

The maximum value in the sample X_n:n is used in the study of extremes, i.e., rainfall, floods, sea level and temperature.
X_1:n is useful for estimation strength of a chain that depends on the weakest link.

In Section 2 and Section 3 of this paper, we give theorems and results related to pf and df of the single order statistic, respectively. Also, we give an example in Section 3. Section 4 presents some conclusions.

2 Theorems for Distribution and Probability Functions

Here, expressions concerning pf and df of X_r:n are demonstrated. Consider

{Xr:n=x}, r=1,2,…,n.

The above event can be expressed as: (r-1-i) X_k's are < x, (i + 1 + j) X_k's are = x and (n - j -r) X_k's are > x with probabilities F(x-), f(x), 1-F(x) and F_k(x-) = P(X_k < x) (i = 0,1,..., r-1, j = 0,1,..., n-r and x = 0,1,2,...), respectively. Therefore, the pf of X_r:n can be denoted as

fr:n(x)=P{Xr:n=x}.

Hence, the above equation is given by

fr:n(x)=∑i=0r−1∑j=0n−r1(r−1−i)!(i+1+j)!(n−j−r)! ·per[F(x-) r−1−i f(x)i+1+j 1−F(x) n−j−r],(1)

where F(x-) = (F₁(x-), F₂(x-),..., F_n(x-))′, f(x) = (f₁(x), f₂(x),..., f_n(x))′ and 1 - F(x) = (1 - F₁(x), 1 - F₂(x),..., 1 - F_n(x))′ are column vectors.

Using expansion of permanent in (1), the following theorem is expressed.

Theorem 2.1

fr:n(x)=∑i=0r−1∑j=0n−r∑ns1,ns2(∏l1=1r−1−iFs1(l1)(x−))·(∏l2=1i+1+jfs2(l2)(x))∏l3=1n−j−r[1−Fs3(l3)(x)],(2)

where ∑ns1,ns2 denotes sum over ⋃l=12sl for which s_ν ∩ s_v = ϖ for υ ≠ v, ∪l=13sl={1, 2, ⋅⋅ ⋅, n}, s1={s1(1),s1(2),, ⋅⋅ ⋅, s1(r−1−i)},s2={s2(1),s2(2),, ⋅⋅ ⋅, s2(i−1−j)},s3={s3(1),s3(2),, ⋅⋅ ⋅, s3(n−j−r)} and n_{s_l} is cardinality of s_l.

Proof. (1) can also be expressed as

fr:n(x)=∑i=0r−1∑j=0n−r1(r−1−i)!(i+1+j)!(n−j−r)!·∑ns1,ns2per[F(x−)r−1−i][s1/·)·per[f(x)i+1+j][s2/·)·per[1−F(x)n−j−r][s3/·),

where A[s_l/·) is the matrix obtained from A by taking rows whose indices are in s_l. From permanent expansion, we get (2).

(1) can be defined as follows.

Theorem 2.2

fr:n(x)=1(r−1)!(n−r)!∑nς1,nς2 ∫Fς2(1)(x−)Fς2(1)(x)per[vr−1][ς1/·) per[1-vn−r][ς3/·) per[dv1][ς2/·),(3)

where v = (v₁, v₂, ..., v_n′, dv = (dv₁, dv₂, ..., dv_n′, vς2w−1(kw)=[vς2(1)−Fς2(1)(x−)]fς2w−1(kw)(x)fς2(1)(x)+Fς2w−1(kw)(x−), ⋃l=13ςl={1,2,…,n}, ςϑ∩ςν=ϕ for ϑ≠v, ς1={ς1(1),ς1(2),⋅⋅⋅,ς1(r−1)},ς2={ς2(1)} and ς3={ς3(1),ς3(2),…,ς3(n−r)}..

Proof. Consider (1). It can be written as

fr:n(x)=∑i=0r−1∑j=0n−r1(r−1−i)!(i+1+j)!(n−j−r)!·∑nτ1,nτ2,nτ3,nτ4per[F(x-)r−1−i][τ1/·)· per[f(x)i][τ2/·) per[f(x)1][τ3/·)per[f(x)j][τ4/·)· per[1-F(x)n-j-r][τ5/·),(4)

where ∪l=15τl={1,2,⋅⋅⋅,n}, τϑ∩τν=ϕ for ϑ≠v,τ1={τ1(1),τ1(2),⋅⋅⋅,τ1(r−1−i)},τ2={τ2(1),τ2(2),⋅⋅⋅,τ2(k)},τ3={τ3(1)},τ4={τ4(1),τ4(2),⋅⋅⋅,τ4(m)} and τ5={τ5(1),τ5(2),…,τ5(n−j−r)}.

(4) can be expressed as

fr:n(x)=∑i=0r−1∑j=0n−r1(r−1−i)!(i+1+j)!(n−j−r)! ·∑nτ1,nτ2,nτ3,nτ4per[F(x-)r−1−i][τ1/·)·per[f(x)i][τ2/·)per[f(x)1][τ3/·)·per[f(x)j][τ4/·)per[1−F(x)n−j−r][τ5/·)·∫01(i+1+j)!i! j!yk(1−y)jdy.

This identity can be written as

fr:n(x)=∑i=0r−1∑j=0n−r1(r−1−i)!i!j!(n−j−r)!·∑nτ1,nτ2,nτ3,nτ4∫01per[F(x-)r−1−i][τ1/·)·per[yf(x)i][τ2/·)per[(1−y)f(x)j][τ4/·)·per[1−F(x)n−j−r][τ5/·)per[f(x)dy1][τ3/·)(5)

In (5), if v = y f(x) + F(x-), (6) is obtained

fr:n(x)=∑i=0r−1∑j=0n−r1(r−1−i)!i!j!(n−j−r)!·∑nτ1,nτ2,nτ3,nτ4∫Fτ3(1)(x−)Fτ3(1)(x)per[F(x-)r−1−i][τ1/·)·per[v−F(x-)i][τ2/·)per[F(x)-vj][τ4/·)·per[1−F(x)n−j−r][τ5/·)per[dv1][τ3/·)(6)

Now, consider the following identity

∑i=0r−11(r−1−i)!i!∑nτ1,nτ2per[Ar−1−i][τ1/·)·per[Bi][τ2/·)per[C1][τ3/·)=1(r−1)!∑nς1per[A+Br−1][ς1/·)per[C1][ς2/·).(7)

Using (7) in (6), we get

fr:n(x)=1(r−1)!(n−r)!∑nς1,nς2 ∫Fς2(1)(x−)Fς2(1)(x)per[F(x-)+v-F(x-)r−1][ς1/·)·per[F(x)-v+1-F(x)n−r][ς3/·)per[dv1][ς2/·),

where ς₁ = τ₁ ∪ τ₂, ς₃ = τ₃ and ς₃ = τ₄ ∪ τ₅. Thus, the proof is completed.

Using expansion of permanent in Theorem 2.2, we establish the following identity.

fr:n(x)=∑nς1,nς2 ∫Fς2(1)(x−)Fς2(1)(x)(∏k1=1r−1vς1(k1))(∏k2=1n−r[1−vς3(k2)])dvς2(1).(8)

Specifically in (3), by taking n = 2, r = 2, v = (v₁, v₂)′, dv = (dv₁, dv₂)′ and vς1(1)=[vς2(1)−Fς2(1)(x−)]fς1(1)(x)fς2(1)(x)+Fς1(1)(x−), the following identity is obtained.

f2:2(x)=∑nς1=1∫Fς2(1)(x−1)Fς2(1)(x)per[v][ς1/⋅)per[dv][ς2/⋅)=∑nς1=1∫Fς2(1)(x−1)Fς2(1)(x)vς1(1)dvς2(1)=∑nς1=1[(vς2(1)22−Fς2(1)(x−)vς2(1)fς1(1)(x)fς2(1)(x))+Fς1(1)(x−)vς2(1)]|Fς2(1)(x−)Fς2(1)(x)=∑nς1=1[Fς2(1)(x)+Fς2(1)(x−)2Fς1(1)(x)−Fς2(1)(x−)fς1(1)(x)+Fς1(1)(x−)fς2(1)(x)=F1(x)+F1(x−)2f2(x)−F1(x−)f2(x)+F2(x−)f1(x)+F2(x)+F2(x−)2f1(x)−F2(x−)f1(x)+F1(x−)f2(x)=F1(x)f2(x)+F2(x)f1(x)−f1(x)f2(x).

We now define the df of X_r:n as follows.

Fr:n(x)=∑z=0x∑i=0r−1∑j=0n−r1(r−1−i)!(i+1+j)!(n−j−r)! ·per[F(z-) r−1−if(z)i+1+j 1−F(z)n−j−r].(9)

From permanent expansion in (9), we establish the following theorem.

Theorem 2.3

Fr:n(x)=∑z=0x∑i=0r−1∑j=0n−r∑ns1,ns2(∏l1=1r−1−iFs1(l1)(z−))·(∏l2=1i+1+jfs2(l2)(z))∏l3=1n−j−r[1−Fs3(l3)(z)].(10)

Proof. It can be written that

Fr:n(x)=∑z=0xfr:n(z)(11)

and using (2) in (11), (10) is obtained.

(9) can also be defined as follows.

Theorem 2.4

Fr:n(x)=1(r−1)!(n−r)!∑nς1,nς2∫0Fς2(1)(x)per[vr−1][ς1/·) ·per[1-vn−r][ς3/·) per[dv1][ς2/·)(12)

Proof. Using (3) in (11), (12) is obtained.

Using expansion of permanent in Theorem 2.4, we establish the following identity.

Fr:n(x)=∑nς1,nς2∫0Fς2(1)(x)(∏k1=1r−1vς1(k1))(∏k2=1n−r[1−vς3(k2)])dvς2(1).(13)

3 Results for Distribution and Probability Functions

Now, results concerning to pf and df of X_1:n and X_n:n are given, respectively.

Result 3.1

f1:n(x)=∑j=0n−1∑ns2(∏l2=11+mfs2(l2)(x))∏l3=1n−m−1[1−Fs3(l3)(x)]=1(n−1)!∑nς2∫Fς2(1)(x−)Fς2(1)(x)per[1−vn−1][ς3/⋅)per[dv1][ς2/⋅)=∑nς2∫Fς2(1)(x−)Fς2(1)(x)(∏k2=1n−1[1−vς3(k2)])dvς2(1).(14)

Proof. In (1), (2), (3) and (8), if r = 1, (14) is obtained.

Result 3.2

f1:n(x)=∑i=0n−1∑ns1(∏l1=1n−1−kFs1(l1)(x−))∏l2=1k+1fs2(l2)(x)=1(n−1)!∑nς1∫Fς2(1)(x−)Fς2(1)(x)per[vn−1][ς1/⋅)per[dv1][ς2/⋅)=∑nς1∫Fς2(1)(x−)Fς2(1)(x)(∏k1=1n−1vς1(k1))dvς2(1).(15)

Proof. In (1), (2), (3) and (8), if r = n, (15) is obtained.

Result 3.3

F1:n(x)=∑z=0x∑j=0n−1∑ns2(∏l2=11+mfs2(l2)(z))∏l3=1n−m−1[1−Fs3(l3)(z)]=1(n−1)!∑nς2∫0Fς2(1)(x)per[1−vn−1][ς3/⋅)per[dv1][ς2/⋅)=∑nς2∫0Fς2(1)(x)(∏k2=1n−1[1−vς3(k2)])dvς2(1).(16)

Proof. In (9), (10), (12) and (13), if r = 1, (16) is obtained.

Result 3.4

F1:n(x)=∑z=0x∑i=0n−1∑ns1(∏l1=1n−1−kFs1(l1)(z−))∏l2=1k+1fs2(l2)(z)=1(n−1)!∑nς1∫0Fς2(1)(x)per[vn−1][ς1/⋅)per[dv1][ς2/⋅)=∑nς1∫0Fς2(1)(x)(∏k1=1n−1vς1(k1))dvς2(1).(17)

Proof. In (9), (10), (12) and (13), if r = n, (17) is obtained.

Example

Suppose that three signal machines, where each one has the probability of sending an incorrect signal with 0.01, are running until they submit an incorrect signal. Let X₁, X₂ and X₃ be the numbers of their correct signals. So, they follow geometric distribution with probability p, which is given by

f(x)=P(X=x)=(1−p)xp, x=0,1,2,… andF(x)=1−(1−p)x+1.

Hence, one can get the following probabilities

P(X1:3>2)=1−F1:3(2)=(1−F(2))3=(0.99)9=0.91352,P(X3:3≤4)=F3:3(4)=(F(4))3=(1−(099)5)3= 000012.

4 Conclusions

We showed that the pf and df of the single order statistic of random variables in the nonidentical discrete case are obtained in integral form. Also, both theoretical and numerical results concerning distributions of extreme order statistics are given.

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Received: 2016-1-13

Accepted: 2016-4-26

Published Online: 2016-6-24

Published in Print: 2016-1-1

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