Research Article

# Further Results on Order Statistics from the Generalized Log Logistic Distribution 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

 Services Related Articles in ASCI Search in Google Scholar View Citation Report Citation Science Alert

 How to cite this article: 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, Adeyemi and Ojo 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 and the, generalized log logistic distribution Adeyemi and Ojo.

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 (1.1)

Letting and It can be easily shown that the pdf of GLL (m1, m2) becomes (1.2)

Note that if m1=m2=1, GLL(m1, m2) becomes the log-logistic distribution. It is symmetric around 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 (1.3)

and (1.4)

Also (1.5)

where and (1.6)

where Adeyemi and Adeyemi and Ojo 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 (2.1)

where  and Proof (2.2)

having used (1.1), (1.3) and (1.5) where (2.3)

Integrating by parts, we have (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 where and (2.5) (2.6)

having used (1.1), (1.3) and (1.5) where (2.7)

Integrating by parts, we have (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  where and (2.9)

Proof (2.10)

where (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 (2.12)

Integrating (2.12) by parts, we have 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 where and (2.13)

Corollary 2.2 For s–r ≥ 2 and t=s+1 where and (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.

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.

Theorem 3.1.For 1 ≤ r < s < t < u ≤ n and a, b, c, d ≥ 1 where (3.1)

Proof (3.2)

where (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 (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 where (3.5)

Proof (3.6)

where 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 (3.7)

By integrating (3.7) by parts, we obtain (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 Where (3.9)

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

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.

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. 