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Further Results on Order Statistics from the Generalized Log Logistic Distribution



Shola Adeyemi and Mathew Oladejo Ojo
 
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ABSTRACT

Further to our earlier results, we derive exact explicit expressions for the triple and quadruple moments of order statistics from the generalized log-logistic distribution

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Shola Adeyemi and Mathew Oladejo Ojo, 2004. Further Results on Order Statistics from the Generalized Log Logistic Distribution. Journal of Applied Sciences, 4: 83-89.

DOI: 10.3923/jas.2004.83.89

URL: https://scialert.net/abstract/?doi=jas.2004.83.89

INTRODUCTION

Recently Adeyemi[1], Adeyemi and Ojo[2] initiated the study into the recurrence relations for moments of order statistics from the generalized log logistic distribution. We have obtained recurrence relations for single and product moments of order statistics from a symmetric, Adeyemi[3] and the, generalized log logistic distribution Adeyemi and Ojo[2].

In this paper, we present further results on our earlier studies by presenting recurrence relations for triple and quadruple moments of order statistics from the generalized log logistic distribution.

The probability density function (pdf) of the GLL (m1, m2) distribution is given by

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(1.1)

Letting Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution and Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution It can be easily shown that the pdf of GLL (m1, m2) becomes

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(1.2)

Note that if m1=m2=1, GLL(m1, m2) becomes the log-logistic distribution. It is symmetric around Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution if m1 = m2, positive skewed if m1 > m2 and negative skewed if m2 > m1.

Let X1:n≤X2:n≤....≤Xn:n denote the order statistics obtained when the n Xi’s are arranged in increasing order of magnitude. We denote

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(1.3)

and

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(1.4)

Also

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(1.5)

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

and

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(1.6)

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

Adeyemi[3] and Adeyemi and Ojo[2] have obtained recurrence relations for and expressions for μr, s:n in both symmetric and general cases respectively.

In this paper, we obtain recurrence relations for and for positive integers m1 and m2.

Recurrence relations for triple moments: Theorem 2.1 for 1 ≤ r < s < t ≤ n - m1 - i and a, b, c ≥ 1

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.1)

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

and

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

Proof

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.2)

having used (1.1), (1.3) and (1.5) where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.3)

Integrating by parts, we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

(2.4)

by putting (2.4) in (2.3) and after simplification, we have the relation (2.1)

Theorem 2.2 For 1 ≤ r < s ≤ n – 1 and a, b, c ≤ 1

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

and

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.5)
Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.6)

having used (1.1), (1.3) and (1.5) where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.7)

Integrating by parts, we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.8)

substracting (2.8) into (2.6) and simplyfying the resulting expression yields the relation (2.5).
Theorem 2.3 For 1 ≤ r < s < t ≤ n and a, b, c ≥ 1

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

and

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.9)

Proof

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.10)

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.11)

having used (1.1), (1.3) and (1.5). Upon writing F(x) = F(x) – F(w) + F(w) and 1-F(x) = F(y) – F(x) + 1 – F(y) and using binomial expansion, we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.12)

Integrating (2.12) by parts, we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

By putting the above expression into (2.10) and after simplification, we have the relation (2.9).

Corollary 2.1 Setting s=r+1, t= r+2 we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

and

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.13)

Corollary 2.2 For s–r ≥ 2 and t=s+1

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

and

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(2.14)

Remark 2.1 In theorems 2.1, 2.2 and 2.3 if m1 = m2 = m we obtain relations for triple moments of order statistics from a symmetric generalized log-logistic distribution studied by Adeyemi[3].

Remark 2.2 In theorems 2.1, 2.2 and 2.3 if m1 = m2 =1 we obtain relations fro triple moments of order statistics from the ordinary log-logistics distribution studied by Ali and Khan[4].

Recurrence relations for quadruple moments
Theorem 3.1.For 1 ≤ r < s < t < u ≤ n and a, b, c, d ≥ 1

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.1)

Proof

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.2)

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.3)

having used (1.1), (1.4) and (1.6). Upon integrating (3.3) by parts writing F(z) = F(z) – F(y) + F(y), F(y) = F(y) –F(x) – F(x) and F(x) = F(x) – F(w) + F(w) and using binomial expansion, we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.4)

Upon substituting (3.4) into (3.2) and simplifying, we have the relation (3.1).

Theorem 3.2. For 1 ≤ r < s < t < u ≤ n and a, b, c, d ≥ 1

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.5)

Proof

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.6)

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

having used (1.1), (1.4) and (1.6). Expressing 1–F(x) as 1 – F(y) + F(y) – F(x) and 1 – F(y) as F(z) – F(y) + 1 – F(z), we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.7)

By integrating (3.7) by parts, we obtain

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.8)

By substituting (3.8) into (3.6) and simplyfying the resulting expression, we obtain the relation (3.5)

Corollary 3.1. Setting s= r+1, t= r+2 and u= r+3, we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

Where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.9)

Corollary 3.2. For s≥r+2, t= s+1 and u= s+2, we have

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution

where

Image for - Further Results on Order Statistics from the Generalized Log Logistic Distribution
(3.10)

Remark 3.1 In theorems 3.1 and 3.2 if we set m1=m2=m we obtain relations for quadruple moments of order statistics from a symmetric generalized log-logistic distribution studied by Adeyemi[3].

Remark 3.1 In theorems 3.1 and 3.2 if we set m1=m2=1 we obtain relations for quadruple moments of order statistics from the ordinary log-logistic distribution studied by Ali and Khan[4].

REFERENCES

1:  Adeyemi, S., 2002. Some recurrence relations for moments of order statistics from a symmetric generalized log logistic distribution. http://interstat.statjournals.net/YEAR/2002/articles/0212001.pdf.

2:  Adeyemi, S., 2002. Some recurrence relations for single and product moments of order statistics from the generalized pareto distribution. J. Stat. Res., 2: 168-179.

3:  Ali, M.M. and A.H. Khan, 1987. On order statistics from the log-logistic distribution. J. Stat. Plann. Inform., 17: 103-108.

4:  Adeyemi, S., 2002. Some recurrence relations for moments of order statistics from a symmetric Generalized Log-logistics distribution. Inter Stat, No. 1.

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