**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 (m_{1}, m_{2}) distribution is given by

Letting
and
It can be easily shown that the pdf of GLL (m_{1}, m_{2}) becomes

Note that if m_{1}=m_{2}=1, GLL(m_{1}, m_{2})
becomes the log-logistic distribution. It is symmetric around
if m_{1} = m_{2}, positive skewed if m_{1} > m_{2}
and negative skewed if m_{2} > m_{1}.

Let X_{1:n}≤X_{2:n}≤....≤X_{n:n} denote the order statistics obtained when the n X_{i}’s are arranged in increasing order of magnitude. We denote

and

Also

where

and

where

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

In this paper, we obtain recurrence relations for and for positive integers m_{1} and m_{2}.

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

where

and

**Proof**

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

Integrating by parts, we have

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

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

Integrating by parts, we have

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

**Proof**

where

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

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

**Corollary 2.2** For s–r ≥ 2 and t=s+1

where

and

**Remark 2.1** In theorems 2.1, 2.2 and 2.3 if m_{1} = m_{2} = 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 m_{1} = m_{2} =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

where

**Proof**

where

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

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

**Proof**

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

By integrating (3.7) by parts, we obtain

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

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

where

**Remark 3.1** In theorems 3.1 and 3.2 if we set m_{1}=m_{2}=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 m_{1}=m_{2}=1 we obtain relations for quadruple moments of order statistics from the ordinary log-logistic distribution studied by Ali and Khan^{[4]}.